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Theorem ptrest 36077
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 17314 . . . 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 5388 . . . . . . . 8 { (∏t𝐹)} ∈ V
3 ptrest.0 . . . . . . . . . 10 (𝜑𝐴𝑉)
4 fvex 6855 . . . . . . . . . . 11 (𝐹𝑢) ∈ V
54rgenw 3068 . . . . . . . . . 10 𝑢𝐴 (𝐹𝑢) ∈ V
6 eqid 2736 . . . . . . . . . . 11 (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) = (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))
76mpoexxg 8008 . . . . . . . . . 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 7841 . . . . . . . . 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 7683 . . . . . . . 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 3143 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝑆𝑊)
15 ixpexg 8860 . . . . . . . 8 (∀𝑘𝐴 𝑆𝑊X𝑘𝐴 𝑆 ∈ V)
1614, 15syl 17 . . . . . . 7 (𝜑X𝑘𝐴 𝑆 ∈ V)
17 restval 17308 . . . . . . 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 6647 . . . . . . . . 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 5892 . . . . . . . 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 6098 . . . . . . . 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 4603 . . . . . . . . . . . . . 14 (𝑥 ∈ { (∏t𝐹)} → 𝑥 = (∏t𝐹))
2423ineq1d 4171 . . . . . . . . . . . . 13 (𝑥 ∈ { (∏t𝐹)} → (𝑥X𝑘𝐴 𝑆) = ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
2524mpteq2ia 5208 . . . . . . . . . . . 12 (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
26 fvex 6855 . . . . . . . . . . . . . 14 (∏t𝐹) ∈ V
2726uniex 7678 . . . . . . . . . . . . 13 (∏t𝐹) ∈ V
2827inex1 5274 . . . . . . . . . . . . 13 ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ∈ V
29 fmptsn 7113 . . . . . . . . . . . . 13 (( (∏t𝐹) ∈ V ∧ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ∈ V) → {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩} = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)))
3027, 28, 29mp2an 690 . . . . . . . . . . . 12 {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩} = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
3125, 30eqtr4i 2767 . . . . . . . . . . 11 (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩}
3231rneqi 5892 . . . . . . . . . 10 ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = ran {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩}
3327rnsnop 6176 . . . . . . . . . 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 7035 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ Top)
37 inss1 4188 . . . . . . . . . . . . . . 15 ( (𝐹𝑘) ∩ 𝑆) ⊆ (𝐹𝑘)
38 eqid 2736 . . . . . . . . . . . . . . . 16 (𝐹𝑘) = (𝐹𝑘)
3938restuni 22513 . . . . . . . . . . . . . . 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 6855 . . . . . . . . . . . . . . . . 17 (𝐹𝑘) ∈ V
4238restin 22517 . . . . . . . . . . . . . . . . 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 4161 . . . . . . . . . . . . . . . . 17 (𝑆 (𝐹𝑘)) = ( (𝐹𝑘) ∩ 𝑆)
4544oveq2i 7368 . . . . . . . . . . . . . . . 16 ((𝐹𝑘) ↾t (𝑆 (𝐹𝑘))) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆))
4643, 45eqtrdi 2792 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆)))
4746unieqd 4879 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆)))
4840, 47eqtr4d 2779 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ( (𝐹𝑘) ∩ 𝑆) = ((𝐹𝑘) ↾t 𝑆))
4948ixpeq2dva 8850 . . . . . . . . . . . 12 (𝜑X𝑘𝐴 ( (𝐹𝑘) ∩ 𝑆) = X𝑘𝐴 ((𝐹𝑘) ↾t 𝑆))
50 ixpin 8861 . . . . . . . . . . . 12 X𝑘𝐴 ( (𝐹𝑘) ∩ 𝑆) = (X𝑘𝐴 (𝐹𝑘) ∩ X𝑘𝐴 𝑆)
51 nfcv 2907 . . . . . . . . . . . . . 14 𝑦 ((𝐹𝑘) ↾t 𝑆)
52 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑘(𝐹𝑦)
53 nfcv 2907 . . . . . . . . . . . . . . . 16 𝑘t
54 nfcsb1v 3880 . . . . . . . . . . . . . . . 16 𝑘𝑦 / 𝑘𝑆
5552, 53, 54nfov 7387 . . . . . . . . . . . . . . 15 𝑘((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
5655nfuni 4872 . . . . . . . . . . . . . 14 𝑘 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
57 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → (𝐹𝑘) = (𝐹𝑦))
58 csbeq1a 3869 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦𝑆 = 𝑦 / 𝑘𝑆)
5957, 58oveq12d 7375 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6059unieqd 4879 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6151, 56, 60cbvixp 8852 . . . . . . . . . . . . 13 X𝑘𝐴 ((𝐹𝑘) ↾t 𝑆) = X𝑦𝐴 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
62 ixpeq2 8849 . . . . . . . . . . . . . 14 (∀𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) → X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = X𝑦𝐴 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
63 ovex 7390 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) ∈ V
64 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑘𝑦
65 eqid 2736 . . . . . . . . . . . . . . . . 17 (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)) = (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))
6664, 55, 59, 65fvmptf 6969 . . . . . . . . . . . . . . . 16 ((𝑦𝐴 ∧ ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) ∈ V) → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6763, 66mpan2 689 . . . . . . . . . . . . . . 15 (𝑦𝐴 → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6867unieqd 4879 . . . . . . . . . . . . . 14 (𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6962, 68mprg 3070 . . . . . . . . . . . . 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 22945 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
743, 35, 73syl2anc 584 . . . . . . . . . . . 12 (𝜑X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
7574ineq1d 4171 . . . . . . . . . . 11 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ X𝑘𝐴 𝑆) = ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
76 resttop 22511 . . . . . . . . . . . . . 14 (((𝐹𝑘) ∈ Top ∧ 𝑆𝑊) → ((𝐹𝑘) ↾t 𝑆) ∈ Top)
7736, 13, 76syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) ∈ Top)
7877fmpttd 7063 . . . . . . . . . . . 12 (𝜑 → (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)):𝐴⟶Top)
79 eqid 2736 . . . . . . . . . . . . 13 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))
8079ptuni 22945 . . . . . . . . . . . 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 4598 . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤 ∈ V
8685elixp 8842 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤X𝑘𝐴 𝑆 ↔ (𝑤 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑤𝑘) ∈ 𝑆))
8786simprbi 497 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤X𝑘𝐴 𝑆 → ∀𝑘𝐴 (𝑤𝑘) ∈ 𝑆)
88 nfcsb1v 3880 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘𝑢 / 𝑘𝑆
8988nfel2 2925 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑤𝑢) ∈ 𝑢 / 𝑘𝑆
90 fveq2 6842 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑢 → (𝑤𝑘) = (𝑤𝑢))
91 csbeq1a 3869 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑢𝑆 = 𝑢 / 𝑘𝑆)
9290, 91eleq12d 2832 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑢 → ((𝑤𝑘) ∈ 𝑆 ↔ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))
9389, 92rspc 3569 . . . . . . . . . . . . . . . . . . . . . 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 654 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)) ↔ ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ ((𝑤𝑢) ∈ 𝑣 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)))
98 elin 3926 . . . . . . . . . . . . . . . . . . . . 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 3926 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ↔ (𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆))
10382eleq2d 2823 . . . . . . . . . . . . . . . . . . . 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 6190 . . . . . . . . . . . . . . . . . . 19 ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) = {𝑤 (∏t𝐹) ∣ (𝑤𝑢) ∈ 𝑣}
110 df-rab 3408 . . . . . . . . . . . . . . . . . . 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 2875 . . . . . . . . . . . . . . . . . 18 {𝑤𝑤X𝑘𝐴 𝑆} = X𝑘𝐴 𝑆
113111, 112ineq12i 4170 . . . . . . . . . . . . . . . . 17 ({𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)} ∩ {𝑤𝑤X𝑘𝐴 𝑆}) = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)
114 inab 4259 . . . . . . . . . . . . . . . . 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 6190 . . . . . . . . . . . . . . . . 17 ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) = {𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∣ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)}
118 df-rab 3408 . . . . . . . . . . . . . . . . 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 3175 . . . . . . . . . . . . 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 4165 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑦 → (𝑣𝑢 / 𝑘𝑆) = (𝑦𝑢 / 𝑘𝑆))
124123imaeq2d 6013 . . . . . . . . . . . . . . 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 3226 . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . 15 𝑦 ∈ V
129128inex1 5274 . . . . . . . . . . . . . 14 (𝑦𝑢 / 𝑘𝑆) ∈ V
130129a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑢𝐴) ∧ 𝑦 ∈ (𝐹𝑢)) → (𝑦𝑢 / 𝑘𝑆) ∈ V)
131 ovex 7390 . . . . . . . . . . . . . . . . 17 ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆) ∈ V
132 nfcv 2907 . . . . . . . . . . . . . . . . . 18 𝑘𝑢
133 nfcv 2907 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐹𝑢)
134133, 53, 88nfov 7387 . . . . . . . . . . . . . . . . . 18 𝑘((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆)
135 fveq2 6842 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑢 → (𝐹𝑘) = (𝐹𝑢))
136135, 91oveq12d 7375 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
137132, 134, 136, 65fvmptf 6969 . . . . . . . . . . . . . . . . 17 ((𝑢𝐴 ∧ ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆) ∈ V) → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
138131, 137mpan2 689 . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . 14 ((𝜑𝑢𝐴) → (𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↔ 𝑣 ∈ ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆)))
141 nfv 1917 . . . . . . . . . . . . . . . . 17 𝑘(𝜑𝑢𝐴)
142 nfcsb1v 3880 . . . . . . . . . . . . . . . . . 18 𝑘𝑢 / 𝑘𝑊
14388, 142nfel 2921 . . . . . . . . . . . . . . . . 17 𝑘𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊
144141, 143nfim 1899 . . . . . . . . . . . . . . . 16 𝑘((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)
145 eleq1w 2820 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 → (𝑘𝐴𝑢𝐴))
146145anbi2d 629 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → ((𝜑𝑘𝐴) ↔ (𝜑𝑢𝐴)))
147 csbeq1a 3869 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢𝑊 = 𝑢 / 𝑘𝑊)
14891, 147eleq12d 2832 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → (𝑆𝑊𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊))
149146, 148imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑢 → (((𝜑𝑘𝐴) → 𝑆𝑊) ↔ ((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)))
150144, 149, 13chvarfv 2233 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)
151 elrest 17309 . . . . . . . . . . . . . . 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 6009 . . . . . . . . . . . . . . 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 5366 . . . . . . . . . . . 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 3173 . . . . . . . . . 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 5910 . . . . . . . . . 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 3268 . . . . . . . . . . 11 𝑥𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))𝑦 = (𝑥X𝑘𝐴 𝑆)
164 nfv 1917 . . . . . . . . . . 11 𝑦𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)
16527mptex 7173 . . . . . . . . . . . . . . . 16 (𝑤 (∏t𝐹) ↦ (𝑤𝑢)) ∈ V
166165cnvex 7862 . . . . . . . . . . . . . . 15 (𝑤 (∏t𝐹) ↦ (𝑤𝑢)) ∈ V
167166imaex 7853 . . . . . . . . . . . . . 14 ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∈ V
168167rgen2w 3069 . . . . . . . . . . . . 13 𝑢𝐴𝑣 ∈ (𝐹𝑢)((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∈ V
169 ineq1 4165 . . . . . . . . . . . . . . 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 7495 . . . . . . . . . . . . 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 3213 . . . . . . . . . . . 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 7489 . . . . . . . . 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 4124 . . . . . . 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 6846 . . . 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 6846 . 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 22952 . . . . 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 7372 . . 3 (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = ((topGen‘(fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))))) ↾t X𝑘𝐴 𝑆))
191 fvex 6855 . . . 4 (fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ∈ V
192 tgrest 22510 . . . 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 22952 . . 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 1541  wcel 2106  {cab 2713  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  csb 3855  cun 3908  cin 3909  wss 3910  {csn 4586  cop 4592   cuni 4865  cmpt 5188  ccnv 5632  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  Xcixp 8835  ficfi 9346  t crest 17302  topGenctg 17319  tcpt 17320  Topctop 22242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-1o 8412  df-er 8648  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9347  df-rest 17304  df-topgen 17325  df-pt 17326  df-top 22243  df-topon 22260  df-bases 22296
This theorem is referenced by:  poimirlem30  36108
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