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Theorem ptrest 35853
Description: Expressing a restriction of a product topology as a product topology. (Contributed by Brendan Leahy, 24-Mar-2019.)
Hypotheses
Ref Expression
ptrest.0 (𝜑𝐴𝑉)
ptrest.1 (𝜑𝐹:𝐴⟶Top)
ptrest.2 ((𝜑𝑘𝐴) → 𝑆𝑊)
Assertion
Ref Expression
ptrest (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))))
Distinct variable groups:   𝜑,𝑘   𝐴,𝑘   𝑘,𝐹   𝑘,𝑉
Allowed substitution hints:   𝑆(𝑘)   𝑊(𝑘)

Proof of Theorem ptrest
Dummy variables 𝑢 𝑣 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 firest 17217 . . . 4 (fi‘(({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↾t X𝑘𝐴 𝑆)) = ((fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ↾t X𝑘𝐴 𝑆)
2 snex 5368 . . . . . . . 8 { (∏t𝐹)} ∈ V
3 ptrest.0 . . . . . . . . . 10 (𝜑𝐴𝑉)
4 fvex 6824 . . . . . . . . . . 11 (𝐹𝑢) ∈ V
54rgenw 3065 . . . . . . . . . 10 𝑢𝐴 (𝐹𝑢) ∈ V
6 eqid 2736 . . . . . . . . . . 11 (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) = (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))
76mpoexxg 7962 . . . . . . . . . 10 ((𝐴𝑉 ∧ ∀𝑢𝐴 (𝐹𝑢) ∈ V) → (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ∈ V)
83, 5, 7sylancl 586 . . . . . . . . 9 (𝜑 → (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ∈ V)
9 rnexg 7797 . . . . . . . . 9 ((𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ∈ V → ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ∈ V)
108, 9syl 17 . . . . . . . 8 (𝜑 → ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ∈ V)
11 unexg 7640 . . . . . . . 8 (({ (∏t𝐹)} ∈ V ∧ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ∈ V) → ({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ∈ V)
122, 10, 11sylancr 587 . . . . . . 7 (𝜑 → ({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ∈ V)
13 ptrest.2 . . . . . . . . 9 ((𝜑𝑘𝐴) → 𝑆𝑊)
1413ralrimiva 3139 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝑆𝑊)
15 ixpexg 8759 . . . . . . . 8 (∀𝑘𝐴 𝑆𝑊X𝑘𝐴 𝑆 ∈ V)
1614, 15syl 17 . . . . . . 7 (𝜑X𝑘𝐴 𝑆 ∈ V)
17 restval 17211 . . . . . . 7 ((({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ∈ V ∧ X𝑘𝐴 𝑆 ∈ V) → (({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↾t X𝑘𝐴 𝑆) = ran (𝑥 ∈ ({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↦ (𝑥X𝑘𝐴 𝑆)))
1812, 16, 17syl2anc 584 . . . . . 6 (𝜑 → (({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↾t X𝑘𝐴 𝑆) = ran (𝑥 ∈ ({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↦ (𝑥X𝑘𝐴 𝑆)))
19 mptun 6616 . . . . . . . . 9 (𝑥 ∈ ({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↦ (𝑥X𝑘𝐴 𝑆)) = ((𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) ∪ (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)))
2019rneqi 5865 . . . . . . . 8 ran (𝑥 ∈ ({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↦ (𝑥X𝑘𝐴 𝑆)) = ran ((𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) ∪ (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)))
21 rnun 6071 . . . . . . . 8 ran ((𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) ∪ (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆))) = (ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) ∪ ran (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)))
2220, 21eqtri 2764 . . . . . . 7 ran (𝑥 ∈ ({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↦ (𝑥X𝑘𝐴 𝑆)) = (ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) ∪ ran (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)))
23 elsni 4587 . . . . . . . . . . . . . 14 (𝑥 ∈ { (∏t𝐹)} → 𝑥 = (∏t𝐹))
2423ineq1d 4155 . . . . . . . . . . . . 13 (𝑥 ∈ { (∏t𝐹)} → (𝑥X𝑘𝐴 𝑆) = ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
2524mpteq2ia 5189 . . . . . . . . . . . 12 (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
26 fvex 6824 . . . . . . . . . . . . . 14 (∏t𝐹) ∈ V
2726uniex 7635 . . . . . . . . . . . . 13 (∏t𝐹) ∈ V
2827inex1 5255 . . . . . . . . . . . . 13 ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ∈ V
29 fmptsn 7078 . . . . . . . . . . . . 13 (( (∏t𝐹) ∈ V ∧ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ∈ V) → {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩} = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)))
3027, 28, 29mp2an 689 . . . . . . . . . . . 12 {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩} = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
3125, 30eqtr4i 2767 . . . . . . . . . . 11 (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩}
3231rneqi 5865 . . . . . . . . . 10 ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = ran {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩}
3327rnsnop 6149 . . . . . . . . . 10 ran {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩} = {( (∏t𝐹) ∩ X𝑘𝐴 𝑆)}
3432, 33eqtri 2764 . . . . . . . . 9 ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = {( (∏t𝐹) ∩ X𝑘𝐴 𝑆)}
35 ptrest.1 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝐴⟶Top)
3635ffvelcdmda 7000 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ Top)
37 inss1 4172 . . . . . . . . . . . . . . 15 ( (𝐹𝑘) ∩ 𝑆) ⊆ (𝐹𝑘)
38 eqid 2736 . . . . . . . . . . . . . . . 16 (𝐹𝑘) = (𝐹𝑘)
3938restuni 22393 . . . . . . . . . . . . . . 15 (((𝐹𝑘) ∈ Top ∧ ( (𝐹𝑘) ∩ 𝑆) ⊆ (𝐹𝑘)) → ( (𝐹𝑘) ∩ 𝑆) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆)))
4036, 37, 39sylancl 586 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ( (𝐹𝑘) ∩ 𝑆) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆)))
41 fvex 6824 . . . . . . . . . . . . . . . . 17 (𝐹𝑘) ∈ V
4238restin 22397 . . . . . . . . . . . . . . . . 17 (((𝐹𝑘) ∈ V ∧ 𝑆𝑊) → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑘) ↾t (𝑆 (𝐹𝑘))))
4341, 13, 42sylancr 587 . . . . . . . . . . . . . . . 16 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑘) ↾t (𝑆 (𝐹𝑘))))
44 incom 4145 . . . . . . . . . . . . . . . . 17 (𝑆 (𝐹𝑘)) = ( (𝐹𝑘) ∩ 𝑆)
4544oveq2i 7327 . . . . . . . . . . . . . . . 16 ((𝐹𝑘) ↾t (𝑆 (𝐹𝑘))) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆))
4643, 45eqtrdi 2792 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆)))
4746unieqd 4863 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆)))
4840, 47eqtr4d 2779 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ( (𝐹𝑘) ∩ 𝑆) = ((𝐹𝑘) ↾t 𝑆))
4948ixpeq2dva 8749 . . . . . . . . . . . 12 (𝜑X𝑘𝐴 ( (𝐹𝑘) ∩ 𝑆) = X𝑘𝐴 ((𝐹𝑘) ↾t 𝑆))
50 ixpin 8760 . . . . . . . . . . . 12 X𝑘𝐴 ( (𝐹𝑘) ∩ 𝑆) = (X𝑘𝐴 (𝐹𝑘) ∩ X𝑘𝐴 𝑆)
51 nfcv 2904 . . . . . . . . . . . . . 14 𝑦 ((𝐹𝑘) ↾t 𝑆)
52 nfcv 2904 . . . . . . . . . . . . . . . 16 𝑘(𝐹𝑦)
53 nfcv 2904 . . . . . . . . . . . . . . . 16 𝑘t
54 nfcsb1v 3866 . . . . . . . . . . . . . . . 16 𝑘𝑦 / 𝑘𝑆
5552, 53, 54nfov 7346 . . . . . . . . . . . . . . 15 𝑘((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
5655nfuni 4856 . . . . . . . . . . . . . 14 𝑘 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
57 fveq2 6811 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → (𝐹𝑘) = (𝐹𝑦))
58 csbeq1a 3855 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦𝑆 = 𝑦 / 𝑘𝑆)
5957, 58oveq12d 7334 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6059unieqd 4863 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6151, 56, 60cbvixp 8751 . . . . . . . . . . . . 13 X𝑘𝐴 ((𝐹𝑘) ↾t 𝑆) = X𝑦𝐴 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
62 ixpeq2 8748 . . . . . . . . . . . . . 14 (∀𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) → X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = X𝑦𝐴 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
63 ovex 7349 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) ∈ V
64 nfcv 2904 . . . . . . . . . . . . . . . . 17 𝑘𝑦
65 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)) = (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))
6664, 55, 59, 65fvmptf 6935 . . . . . . . . . . . . . . . 16 ((𝑦𝐴 ∧ ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) ∈ V) → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6763, 66mpan2 688 . . . . . . . . . . . . . . 15 (𝑦𝐴 → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6867unieqd 4863 . . . . . . . . . . . . . 14 (𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6962, 68mprg 3067 . . . . . . . . . . . . 13 X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = X𝑦𝐴 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
7061, 69eqtr4i 2767 . . . . . . . . . . . 12 X𝑘𝐴 ((𝐹𝑘) ↾t 𝑆) = X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦)
7149, 50, 703eqtr3g 2799 . . . . . . . . . . 11 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ X𝑘𝐴 𝑆) = X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦))
72 eqid 2736 . . . . . . . . . . . . . 14 (∏t𝐹) = (∏t𝐹)
7372ptuni 22825 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
743, 35, 73syl2anc 584 . . . . . . . . . . . 12 (𝜑X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
7574ineq1d 4155 . . . . . . . . . . 11 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ X𝑘𝐴 𝑆) = ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
76 resttop 22391 . . . . . . . . . . . . . 14 (((𝐹𝑘) ∈ Top ∧ 𝑆𝑊) → ((𝐹𝑘) ↾t 𝑆) ∈ Top)
7736, 13, 76syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) ∈ Top)
7877fmpttd 7028 . . . . . . . . . . . 12 (𝜑 → (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)):𝐴⟶Top)
79 eqid 2736 . . . . . . . . . . . . 13 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))
8079ptuni 22825 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)):𝐴⟶Top) → X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))))
813, 78, 80syl2anc 584 . . . . . . . . . . 11 (𝜑X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))))
8271, 75, 813eqtr3d 2784 . . . . . . . . . 10 (𝜑 → ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))))
8382sneqd 4582 . . . . . . . . 9 (𝜑 → {( (∏t𝐹) ∩ X𝑘𝐴 𝑆)} = { (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))})
8434, 83eqtrid 2788 . . . . . . . 8 (𝜑 → ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = { (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))})
85 vex 3444 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤 ∈ V
8685elixp 8741 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤X𝑘𝐴 𝑆 ↔ (𝑤 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑤𝑘) ∈ 𝑆))
8786simprbi 497 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤X𝑘𝐴 𝑆 → ∀𝑘𝐴 (𝑤𝑘) ∈ 𝑆)
88 nfcsb1v 3866 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘𝑢 / 𝑘𝑆
8988nfel2 2922 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑤𝑢) ∈ 𝑢 / 𝑘𝑆
90 fveq2 6811 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑢 → (𝑤𝑘) = (𝑤𝑢))
91 csbeq1a 3855 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑢𝑆 = 𝑢 / 𝑘𝑆)
9290, 91eleq12d 2831 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑢 → ((𝑤𝑘) ∈ 𝑆 ↔ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))
9389, 92rspc 3557 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢𝐴 → (∀𝑘𝐴 (𝑤𝑘) ∈ 𝑆 → (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))
9487, 93syl5 34 . . . . . . . . . . . . . . . . . . . . 21 (𝑢𝐴 → (𝑤X𝑘𝐴 𝑆 → (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))
9594pm4.71d 562 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝐴 → (𝑤X𝑘𝐴 𝑆 ↔ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)))
9695anbi2d 629 . . . . . . . . . . . . . . . . . . 19 (𝑢𝐴 → (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆) ↔ ((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))))
97 an4 653 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)) ↔ ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ ((𝑤𝑢) ∈ 𝑣 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)))
98 elin 3912 . . . . . . . . . . . . . . . . . . . . 21 ((𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆) ↔ ((𝑤𝑢) ∈ 𝑣 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))
9998anbi2i 623 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)) ↔ ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ ((𝑤𝑢) ∈ 𝑣 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)))
10097, 99bitr4i 277 . . . . . . . . . . . . . . . . . . 19 (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)) ↔ ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)))
10196, 100bitrdi 286 . . . . . . . . . . . . . . . . . 18 (𝑢𝐴 → (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆) ↔ ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))))
102 elin 3912 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ↔ (𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆))
10382eleq2d 2822 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑤 ∈ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ↔ 𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))))
104102, 103bitr3id 284 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ↔ 𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))))
105104anbi1d 630 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)) ↔ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))))
106101, 105sylan9bbr 511 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢𝐴) → (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆) ↔ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))))
107106abbidv 2805 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝐴) → {𝑤 ∣ ((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆)} = {𝑤 ∣ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))})
108 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑤 (∏t𝐹) ↦ (𝑤𝑢)) = (𝑤 (∏t𝐹) ↦ (𝑤𝑢))
109108mptpreima 6163 . . . . . . . . . . . . . . . . . . 19 ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) = {𝑤 (∏t𝐹) ∣ (𝑤𝑢) ∈ 𝑣}
110 df-rab 3404 . . . . . . . . . . . . . . . . . . 19 {𝑤 (∏t𝐹) ∣ (𝑤𝑢) ∈ 𝑣} = {𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)}
111109, 110eqtr2i 2765 . . . . . . . . . . . . . . . . . 18 {𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)} = ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)
112 abid2 2879 . . . . . . . . . . . . . . . . . 18 {𝑤𝑤X𝑘𝐴 𝑆} = X𝑘𝐴 𝑆
113111, 112ineq12i 4154 . . . . . . . . . . . . . . . . 17 ({𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)} ∩ {𝑤𝑤X𝑘𝐴 𝑆}) = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)
114 inab 4243 . . . . . . . . . . . . . . . . 17 ({𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)} ∩ {𝑤𝑤X𝑘𝐴 𝑆}) = {𝑤 ∣ ((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆)}
115113, 114eqtr3i 2766 . . . . . . . . . . . . . . . 16 (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) = {𝑤 ∣ ((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆)}
116 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) = (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢))
117116mptpreima 6163 . . . . . . . . . . . . . . . . 17 ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) = {𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∣ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)}
118 df-rab 3404 . . . . . . . . . . . . . . . . 17 {𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∣ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)} = {𝑤 ∣ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))}
119117, 118eqtri 2764 . . . . . . . . . . . . . . . 16 ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) = {𝑤 ∣ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))}
120107, 115, 1193eqtr4g 2801 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐴) → (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)))
121120eqeq2d 2747 . . . . . . . . . . . . . 14 ((𝜑𝑢𝐴) → (𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ 𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆))))
122121rexbidv 3171 . . . . . . . . . . . . 13 ((𝜑𝑢𝐴) → (∃𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ ∃𝑣 ∈ (𝐹𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆))))
123 ineq1 4149 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑦 → (𝑣𝑢 / 𝑘𝑆) = (𝑦𝑢 / 𝑘𝑆))
124123imaeq2d 5986 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆)))
125124eqeq2d 2747 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) ↔ 𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆))))
126125cbvrexvw 3222 . . . . . . . . . . . . 13 (∃𝑣 ∈ (𝐹𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) ↔ ∃𝑦 ∈ (𝐹𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆)))
127122, 126bitrdi 286 . . . . . . . . . . . 12 ((𝜑𝑢𝐴) → (∃𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ ∃𝑦 ∈ (𝐹𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆))))
128 vex 3444 . . . . . . . . . . . . . . 15 𝑦 ∈ V
129128inex1 5255 . . . . . . . . . . . . . 14 (𝑦𝑢 / 𝑘𝑆) ∈ V
130129a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑢𝐴) ∧ 𝑦 ∈ (𝐹𝑢)) → (𝑦𝑢 / 𝑘𝑆) ∈ V)
131 ovex 7349 . . . . . . . . . . . . . . . . 17 ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆) ∈ V
132 nfcv 2904 . . . . . . . . . . . . . . . . . 18 𝑘𝑢
133 nfcv 2904 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐹𝑢)
134133, 53, 88nfov 7346 . . . . . . . . . . . . . . . . . 18 𝑘((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆)
135 fveq2 6811 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑢 → (𝐹𝑘) = (𝐹𝑢))
136135, 91oveq12d 7334 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
137132, 134, 136, 65fvmptf 6935 . . . . . . . . . . . . . . . . 17 ((𝑢𝐴 ∧ ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆) ∈ V) → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
138131, 137mpan2 688 . . . . . . . . . . . . . . . 16 (𝑢𝐴 → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
139138adantl 482 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐴) → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
140139eleq2d 2822 . . . . . . . . . . . . . 14 ((𝜑𝑢𝐴) → (𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↔ 𝑣 ∈ ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆)))
141 nfv 1916 . . . . . . . . . . . . . . . . 17 𝑘(𝜑𝑢𝐴)
142 nfcsb1v 3866 . . . . . . . . . . . . . . . . . 18 𝑘𝑢 / 𝑘𝑊
14388, 142nfel 2918 . . . . . . . . . . . . . . . . 17 𝑘𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊
144141, 143nfim 1898 . . . . . . . . . . . . . . . 16 𝑘((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)
145 eleq1w 2819 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 → (𝑘𝐴𝑢𝐴))
146145anbi2d 629 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → ((𝜑𝑘𝐴) ↔ (𝜑𝑢𝐴)))
147 csbeq1a 3855 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢𝑊 = 𝑢 / 𝑘𝑊)
14891, 147eleq12d 2831 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → (𝑆𝑊𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊))
149146, 148imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑢 → (((𝜑𝑘𝐴) → 𝑆𝑊) ↔ ((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)))
150144, 149, 13chvarfv 2232 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)
151 elrest 17212 . . . . . . . . . . . . . . 15 (((𝐹𝑢) ∈ V ∧ 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊) → (𝑣 ∈ ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆) ↔ ∃𝑦 ∈ (𝐹𝑢)𝑣 = (𝑦𝑢 / 𝑘𝑆)))
1524, 150, 151sylancr 587 . . . . . . . . . . . . . 14 ((𝜑𝑢𝐴) → (𝑣 ∈ ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆) ↔ ∃𝑦 ∈ (𝐹𝑢)𝑣 = (𝑦𝑢 / 𝑘𝑆)))
153140, 152bitrd 278 . . . . . . . . . . . . 13 ((𝜑𝑢𝐴) → (𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↔ ∃𝑦 ∈ (𝐹𝑢)𝑣 = (𝑦𝑢 / 𝑘𝑆)))
154 imaeq2 5982 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦𝑢 / 𝑘𝑆) → ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣) = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆)))
155154eqeq2d 2747 . . . . . . . . . . . . . 14 (𝑣 = (𝑦𝑢 / 𝑘𝑆) → (𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣) ↔ 𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆))))
156155adantl 482 . . . . . . . . . . . . 13 (((𝜑𝑢𝐴) ∧ 𝑣 = (𝑦𝑢 / 𝑘𝑆)) → (𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣) ↔ 𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆))))
157130, 153, 156rexxfr2d 5348 . . . . . . . . . . . 12 ((𝜑𝑢𝐴) → (∃𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣) ↔ ∃𝑦 ∈ (𝐹𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆))))
158127, 157bitr4d 281 . . . . . . . . . . 11 ((𝜑𝑢𝐴) → (∃𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ ∃𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)))
159158rexbidva 3169 . . . . . . . . . 10 (𝜑 → (∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ ∃𝑢𝐴𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)))
160159abbidv 2805 . . . . . . . . 9 (𝜑 → {𝑥 ∣ ∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)} = {𝑥 ∣ ∃𝑢𝐴𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)})
161 eqid 2736 . . . . . . . . . . 11 (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)) = (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆))
162161rnmpt 5883 . . . . . . . . . 10 ran (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)) = {𝑦 ∣ ∃𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))𝑦 = (𝑥X𝑘𝐴 𝑆)}
163 nfre1 3264 . . . . . . . . . . 11 𝑥𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))𝑦 = (𝑥X𝑘𝐴 𝑆)
164 nfv 1916 . . . . . . . . . . 11 𝑦𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)
16527mptex 7138 . . . . . . . . . . . . . . . 16 (𝑤 (∏t𝐹) ↦ (𝑤𝑢)) ∈ V
166165cnvex 7818 . . . . . . . . . . . . . . 15 (𝑤 (∏t𝐹) ↦ (𝑤𝑢)) ∈ V
167166imaex 7809 . . . . . . . . . . . . . 14 ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∈ V
168167rgen2w 3066 . . . . . . . . . . . . 13 𝑢𝐴𝑣 ∈ (𝐹𝑢)((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∈ V
169 ineq1 4149 . . . . . . . . . . . . . . 15 (𝑥 = ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) → (𝑥X𝑘𝐴 𝑆) = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆))
170169eqeq2d 2747 . . . . . . . . . . . . . 14 (𝑥 = ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) → (𝑦 = (𝑥X𝑘𝐴 𝑆) ↔ 𝑦 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)))
1716, 170rexrnmpo 7454 . . . . . . . . . . . . 13 (∀𝑢𝐴𝑣 ∈ (𝐹𝑢)((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∈ V → (∃𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))𝑦 = (𝑥X𝑘𝐴 𝑆) ↔ ∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑦 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)))
172168, 171ax-mp 5 . . . . . . . . . . . 12 (∃𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))𝑦 = (𝑥X𝑘𝐴 𝑆) ↔ ∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑦 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆))
173 eqeq1 2740 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ 𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)))
1741732rexbidv 3209 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑦 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ ∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)))
175172, 174bitrid 282 . . . . . . . . . . 11 (𝑦 = 𝑥 → (∃𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))𝑦 = (𝑥X𝑘𝐴 𝑆) ↔ ∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)))
176163, 164, 175cbvabw 2810 . . . . . . . . . 10 {𝑦 ∣ ∃𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))𝑦 = (𝑥X𝑘𝐴 𝑆)} = {𝑥 ∣ ∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)}
177162, 176eqtri 2764 . . . . . . . . 9 ran (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)) = {𝑥 ∣ ∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)}
178 eqid 2736 . . . . . . . . . 10 (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)) = (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣))
179178rnmpo 7448 . . . . . . . . 9 ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)) = {𝑥 ∣ ∃𝑢𝐴𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢)𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)}
180160, 177, 1793eqtr4g 2801 . . . . . . . 8 (𝜑 → ran (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)) = ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)))
18184, 180uneq12d 4108 . . . . . . 7 (𝜑 → (ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) ∪ ran (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆))) = ({ (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))} ∪ ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣))))
18222, 181eqtrid 2788 . . . . . 6 (𝜑 → ran (𝑥 ∈ ({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↦ (𝑥X𝑘𝐴 𝑆)) = ({ (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))} ∪ ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣))))
18318, 182eqtrd 2776 . . . . 5 (𝜑 → (({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↾t X𝑘𝐴 𝑆) = ({ (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))} ∪ ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣))))
184183fveq2d 6815 . . . 4 (𝜑 → (fi‘(({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))) ↾t X𝑘𝐴 𝑆)) = (fi‘({ (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))} ∪ ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)))))
1851, 184eqtr3id 2790 . . 3 (𝜑 → ((fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ↾t X𝑘𝐴 𝑆) = (fi‘({ (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))} ∪ ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣)))))
186185fveq2d 6815 . 2 (𝜑 → (topGen‘((fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ↾t X𝑘𝐴 𝑆)) = (topGen‘(fi‘({ (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))} ∪ ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣))))))
187 eqid 2736 . . . . . 6 (∏t𝐹) = (∏t𝐹)
18872, 187, 6ptval2 22832 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) = (topGen‘(fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))))))
1893, 35, 188syl2anc 584 . . . 4 (𝜑 → (∏t𝐹) = (topGen‘(fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))))))
190189oveq1d 7331 . . 3 (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = ((topGen‘(fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))))) ↾t X𝑘𝐴 𝑆))
191 fvex 6824 . . . 4 (fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ∈ V
192 tgrest 22390 . . . 4 (((fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ∈ V ∧ X𝑘𝐴 𝑆 ∈ V) → (topGen‘((fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ↾t X𝑘𝐴 𝑆)) = ((topGen‘(fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))))) ↾t X𝑘𝐴 𝑆))
193191, 16, 192sylancr 587 . . 3 (𝜑 → (topGen‘((fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ↾t X𝑘𝐴 𝑆)) = ((topGen‘(fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))))) ↾t X𝑘𝐴 𝑆))
194190, 193eqtr4d 2779 . 2 (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = (topGen‘((fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ↾t X𝑘𝐴 𝑆)))
195 eqid 2736 . . . 4 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))
19679, 195, 178ptval2 22832 . . 3 ((𝐴𝑉 ∧ (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)):𝐴⟶Top) → (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) = (topGen‘(fi‘({ (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))} ∪ ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣))))))
1973, 78, 196syl2anc 584 . 2 (𝜑 → (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) = (topGen‘(fi‘({ (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))} ∪ ran (𝑢𝐴, 𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↦ ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣))))))
198186, 194, 1973eqtr4d 2786 1 (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  {cab 2713  wral 3061  wrex 3070  {crab 3403  Vcvv 3440  csb 3841  cun 3894  cin 3895  wss 3896  {csn 4570  cop 4576   cuni 4849  cmpt 5169  ccnv 5606  ran crn 5608  cima 5610   Fn wfn 6460  wf 6461  cfv 6465  (class class class)co 7316  cmpo 7318  Xcixp 8734  ficfi 9245  t crest 17205  topGenctg 17222  tcpt 17223  Topctop 22122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5223  ax-sep 5237  ax-nul 5244  ax-pow 5302  ax-pr 5366  ax-un 7629
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3442  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-pss 3915  df-nul 4267  df-if 4471  df-pw 4546  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4850  df-int 4892  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5170  df-tr 5204  df-id 5506  df-eprel 5512  df-po 5520  df-so 5521  df-fr 5562  df-we 5564  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-ord 6291  df-on 6292  df-lim 6293  df-suc 6294  df-iota 6417  df-fun 6467  df-fn 6468  df-f 6469  df-f1 6470  df-fo 6471  df-f1o 6472  df-fv 6473  df-ov 7319  df-oprab 7320  df-mpo 7321  df-om 7759  df-1st 7877  df-2nd 7878  df-1o 8345  df-er 8547  df-ixp 8735  df-en 8783  df-dom 8784  df-fin 8786  df-fi 9246  df-rest 17207  df-topgen 17228  df-pt 17229  df-top 22123  df-topon 22140  df-bases 22176
This theorem is referenced by:  poimirlem30  35884
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