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Theorem ptrest 37606
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 17479 . . . 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 5442 . . . . . . . 8 { (∏t𝐹)} ∈ V
3 ptrest.0 . . . . . . . . . 10 (𝜑𝐴𝑉)
4 fvex 6920 . . . . . . . . . . 11 (𝐹𝑢) ∈ V
54rgenw 3063 . . . . . . . . . 10 𝑢𝐴 (𝐹𝑢) ∈ V
6 eqid 2735 . . . . . . . . . . 11 (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) = (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))
76mpoexxg 8099 . . . . . . . . . 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 7925 . . . . . . . . 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 7762 . . . . . . . 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 3144 . . . . . . . 8 (𝜑 → ∀𝑘𝐴 𝑆𝑊)
15 ixpexg 8961 . . . . . . . 8 (∀𝑘𝐴 𝑆𝑊X𝑘𝐴 𝑆 ∈ V)
1614, 15syl 17 . . . . . . 7 (𝜑X𝑘𝐴 𝑆 ∈ V)
17 restval 17473 . . . . . . 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 6715 . . . . . . . . 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 5951 . . . . . . . 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 6168 . . . . . . . 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 2763 . . . . . . 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 4648 . . . . . . . . . . . . . 14 (𝑥 ∈ { (∏t𝐹)} → 𝑥 = (∏t𝐹))
2423ineq1d 4227 . . . . . . . . . . . . 13 (𝑥 ∈ { (∏t𝐹)} → (𝑥X𝑘𝐴 𝑆) = ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
2524mpteq2ia 5251 . . . . . . . . . . . 12 (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
26 fvex 6920 . . . . . . . . . . . . . 14 (∏t𝐹) ∈ V
2726uniex 7760 . . . . . . . . . . . . 13 (∏t𝐹) ∈ V
2827inex1 5323 . . . . . . . . . . . . 13 ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ∈ V
29 fmptsn 7187 . . . . . . . . . . . . 13 (( (∏t𝐹) ∈ V ∧ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ∈ V) → {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩} = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)))
3027, 28, 29mp2an 692 . . . . . . . . . . . 12 {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩} = (𝑥 ∈ { (∏t𝐹)} ↦ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
3125, 30eqtr4i 2766 . . . . . . . . . . 11 (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩}
3231rneqi 5951 . . . . . . . . . 10 ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = ran {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩}
3327rnsnop 6246 . . . . . . . . . 10 ran {⟨ (∏t𝐹), ( (∏t𝐹) ∩ X𝑘𝐴 𝑆)⟩} = {( (∏t𝐹) ∩ X𝑘𝐴 𝑆)}
3432, 33eqtri 2763 . . . . . . . . 9 ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = {( (∏t𝐹) ∩ X𝑘𝐴 𝑆)}
35 ptrest.1 . . . . . . . . . . . . . . . 16 (𝜑𝐹:𝐴⟶Top)
3635ffvelcdmda 7104 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → (𝐹𝑘) ∈ Top)
37 inss1 4245 . . . . . . . . . . . . . . 15 ( (𝐹𝑘) ∩ 𝑆) ⊆ (𝐹𝑘)
38 eqid 2735 . . . . . . . . . . . . . . . 16 (𝐹𝑘) = (𝐹𝑘)
3938restuni 23186 . . . . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . . . . 17 (𝐹𝑘) ∈ V
4238restin 23190 . . . . . . . . . . . . . . . . 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 4217 . . . . . . . . . . . . . . . . 17 (𝑆 (𝐹𝑘)) = ( (𝐹𝑘) ∩ 𝑆)
4544oveq2i 7442 . . . . . . . . . . . . . . . 16 ((𝐹𝑘) ↾t (𝑆 (𝐹𝑘))) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆))
4643, 45eqtrdi 2791 . . . . . . . . . . . . . . 15 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆)))
4746unieqd 4925 . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑘) ↾t ( (𝐹𝑘) ∩ 𝑆)))
4840, 47eqtr4d 2778 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ( (𝐹𝑘) ∩ 𝑆) = ((𝐹𝑘) ↾t 𝑆))
4948ixpeq2dva 8951 . . . . . . . . . . . 12 (𝜑X𝑘𝐴 ( (𝐹𝑘) ∩ 𝑆) = X𝑘𝐴 ((𝐹𝑘) ↾t 𝑆))
50 ixpin 8962 . . . . . . . . . . . 12 X𝑘𝐴 ( (𝐹𝑘) ∩ 𝑆) = (X𝑘𝐴 (𝐹𝑘) ∩ X𝑘𝐴 𝑆)
51 nfcv 2903 . . . . . . . . . . . . . 14 𝑦 ((𝐹𝑘) ↾t 𝑆)
52 nfcv 2903 . . . . . . . . . . . . . . . 16 𝑘(𝐹𝑦)
53 nfcv 2903 . . . . . . . . . . . . . . . 16 𝑘t
54 nfcsb1v 3933 . . . . . . . . . . . . . . . 16 𝑘𝑦 / 𝑘𝑆
5552, 53, 54nfov 7461 . . . . . . . . . . . . . . 15 𝑘((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
5655nfuni 4919 . . . . . . . . . . . . . 14 𝑘 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
57 fveq2 6907 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦 → (𝐹𝑘) = (𝐹𝑦))
58 csbeq1a 3922 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑦𝑆 = 𝑦 / 𝑘𝑆)
5957, 58oveq12d 7449 . . . . . . . . . . . . . . 15 (𝑘 = 𝑦 → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6059unieqd 4925 . . . . . . . . . . . . . 14 (𝑘 = 𝑦 ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6151, 56, 60cbvixp 8953 . . . . . . . . . . . . 13 X𝑘𝐴 ((𝐹𝑘) ↾t 𝑆) = X𝑦𝐴 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
62 ixpeq2 8950 . . . . . . . . . . . . . 14 (∀𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) → X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = X𝑦𝐴 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
63 ovex 7464 . . . . . . . . . . . . . . . 16 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) ∈ V
64 nfcv 2903 . . . . . . . . . . . . . . . . 17 𝑘𝑦
65 eqid 2735 . . . . . . . . . . . . . . . . 17 (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)) = (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))
6664, 55, 59, 65fvmptf 7037 . . . . . . . . . . . . . . . 16 ((𝑦𝐴 ∧ ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆) ∈ V) → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6763, 66mpan2 691 . . . . . . . . . . . . . . 15 (𝑦𝐴 → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6867unieqd 4925 . . . . . . . . . . . . . 14 (𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆))
6962, 68mprg 3065 . . . . . . . . . . . . 13 X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦) = X𝑦𝐴 ((𝐹𝑦) ↾t 𝑦 / 𝑘𝑆)
7061, 69eqtr4i 2766 . . . . . . . . . . . 12 X𝑘𝐴 ((𝐹𝑘) ↾t 𝑆) = X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦)
7149, 50, 703eqtr3g 2798 . . . . . . . . . . 11 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ X𝑘𝐴 𝑆) = X𝑦𝐴 ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑦))
72 eqid 2735 . . . . . . . . . . . . . 14 (∏t𝐹) = (∏t𝐹)
7372ptuni 23618 . . . . . . . . . . . . 13 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
743, 35, 73syl2anc 584 . . . . . . . . . . . 12 (𝜑X𝑘𝐴 (𝐹𝑘) = (∏t𝐹))
7574ineq1d 4227 . . . . . . . . . . 11 (𝜑 → (X𝑘𝐴 (𝐹𝑘) ∩ X𝑘𝐴 𝑆) = ( (∏t𝐹) ∩ X𝑘𝐴 𝑆))
76 resttop 23184 . . . . . . . . . . . . . 14 (((𝐹𝑘) ∈ Top ∧ 𝑆𝑊) → ((𝐹𝑘) ↾t 𝑆) ∈ Top)
7736, 13, 76syl2anc 584 . . . . . . . . . . . . 13 ((𝜑𝑘𝐴) → ((𝐹𝑘) ↾t 𝑆) ∈ Top)
7877fmpttd 7135 . . . . . . . . . . . 12 (𝜑 → (𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)):𝐴⟶Top)
79 eqid 2735 . . . . . . . . . . . . 13 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))
8079ptuni 23618 . . . . . . . . . . . 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 2783 . . . . . . . . . 10 (𝜑 → ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))))
8382sneqd 4643 . . . . . . . . 9 (𝜑 → {( (∏t𝐹) ∩ X𝑘𝐴 𝑆)} = { (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))})
8434, 83eqtrid 2787 . . . . . . . 8 (𝜑 → ran (𝑥 ∈ { (∏t𝐹)} ↦ (𝑥X𝑘𝐴 𝑆)) = { (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))})
85 vex 3482 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑤 ∈ V
8685elixp 8943 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤X𝑘𝐴 𝑆 ↔ (𝑤 Fn 𝐴 ∧ ∀𝑘𝐴 (𝑤𝑘) ∈ 𝑆))
8786simprbi 496 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤X𝑘𝐴 𝑆 → ∀𝑘𝐴 (𝑤𝑘) ∈ 𝑆)
88 nfcsb1v 3933 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑘𝑢 / 𝑘𝑆
8988nfel2 2922 . . . . . . . . . . . . . . . . . . . . . . 23 𝑘(𝑤𝑢) ∈ 𝑢 / 𝑘𝑆
90 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑢 → (𝑤𝑘) = (𝑤𝑢))
91 csbeq1a 3922 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑘 = 𝑢𝑆 = 𝑢 / 𝑘𝑆)
9290, 91eleq12d 2833 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑘 = 𝑢 → ((𝑤𝑘) ∈ 𝑆 ↔ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))
9389, 92rspc 3610 . . . . . . . . . . . . . . . . . . . . . 22 (𝑢𝐴 → (∀𝑘𝐴 (𝑤𝑘) ∈ 𝑆 → (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))
9487, 93syl5 34 . . . . . . . . . . . . . . . . . . . . 21 (𝑢𝐴 → (𝑤X𝑘𝐴 𝑆 → (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))
9594pm4.71d 561 . . . . . . . . . . . . . . . . . . . 20 (𝑢𝐴 → (𝑤X𝑘𝐴 𝑆 ↔ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)))
9695anbi2d 630 . . . . . . . . . . . . . . . . . . 19 (𝑢𝐴 → (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆) ↔ ((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆))))
97 an4 656 . . . . . . . . . . . . . . . . . . . 20 (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)) ↔ ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ ((𝑤𝑢) ∈ 𝑣 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)))
98 elin 3979 . . . . . . . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . . . . . . 19 (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ (𝑤X𝑘𝐴 𝑆 ∧ (𝑤𝑢) ∈ 𝑢 / 𝑘𝑆)) ↔ ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)))
10196, 100bitrdi 287 . . . . . . . . . . . . . . . . . 18 (𝑢𝐴 → (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆) ↔ ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))))
102 elin 3979 . . . . . . . . . . . . . . . . . . . 20 (𝑤 ∈ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ↔ (𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆))
10382eleq2d 2825 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑤 ∈ ( (∏t𝐹) ∩ X𝑘𝐴 𝑆) ↔ 𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))))
104102, 103bitr3id 285 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ↔ 𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))))
105104anbi1d 631 . . . . . . . . . . . . . . . . . 18 (𝜑 → (((𝑤 (∏t𝐹) ∧ 𝑤X𝑘𝐴 𝑆) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)) ↔ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))))
106101, 105sylan9bbr 510 . . . . . . . . . . . . . . . . 17 ((𝜑𝑢𝐴) → (((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆) ↔ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))))
107106abbidv 2806 . . . . . . . . . . . . . . . 16 ((𝜑𝑢𝐴) → {𝑤 ∣ ((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆)} = {𝑤 ∣ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))})
108 eqid 2735 . . . . . . . . . . . . . . . . . . . 20 (𝑤 (∏t𝐹) ↦ (𝑤𝑢)) = (𝑤 (∏t𝐹) ↦ (𝑤𝑢))
109108mptpreima 6260 . . . . . . . . . . . . . . . . . . 19 ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) = {𝑤 (∏t𝐹) ∣ (𝑤𝑢) ∈ 𝑣}
110 df-rab 3434 . . . . . . . . . . . . . . . . . . 19 {𝑤 (∏t𝐹) ∣ (𝑤𝑢) ∈ 𝑣} = {𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)}
111109, 110eqtr2i 2764 . . . . . . . . . . . . . . . . . 18 {𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)} = ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)
112 abid2 2877 . . . . . . . . . . . . . . . . . 18 {𝑤𝑤X𝑘𝐴 𝑆} = X𝑘𝐴 𝑆
113111, 112ineq12i 4226 . . . . . . . . . . . . . . . . 17 ({𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)} ∩ {𝑤𝑤X𝑘𝐴 𝑆}) = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)
114 inab 4315 . . . . . . . . . . . . . . . . 17 ({𝑤 ∣ (𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣)} ∩ {𝑤𝑤X𝑘𝐴 𝑆}) = {𝑤 ∣ ((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆)}
115113, 114eqtr3i 2765 . . . . . . . . . . . . . . . 16 (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) = {𝑤 ∣ ((𝑤 (∏t𝐹) ∧ (𝑤𝑢) ∈ 𝑣) ∧ 𝑤X𝑘𝐴 𝑆)}
116 eqid 2735 . . . . . . . . . . . . . . . . . 18 (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) = (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢))
117116mptpreima 6260 . . . . . . . . . . . . . . . . 17 ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) = {𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∣ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)}
118 df-rab 3434 . . . . . . . . . . . . . . . . 17 {𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∣ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆)} = {𝑤 ∣ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))}
119117, 118eqtri 2763 . . . . . . . . . . . . . . . 16 ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) = {𝑤 ∣ (𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ∧ (𝑤𝑢) ∈ (𝑣𝑢 / 𝑘𝑆))}
120107, 115, 1193eqtr4g 2800 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐴) → (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)))
121120eqeq2d 2746 . . . . . . . . . . . . . 14 ((𝜑𝑢𝐴) → (𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ 𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆))))
122121rexbidv 3177 . . . . . . . . . . . . 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 4221 . . . . . . . . . . . . . . . 16 (𝑣 = 𝑦 → (𝑣𝑢 / 𝑘𝑆) = (𝑦𝑢 / 𝑘𝑆))
124123imaeq2d 6080 . . . . . . . . . . . . . . 15 (𝑣 = 𝑦 → ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆)))
125124eqeq2d 2746 . . . . . . . . . . . . . 14 (𝑣 = 𝑦 → (𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑣𝑢 / 𝑘𝑆)) ↔ 𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆))))
126125cbvrexvw 3236 . . . . . . . . . . . . 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 287 . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . 15 𝑦 ∈ V
129128inex1 5323 . . . . . . . . . . . . . 14 (𝑦𝑢 / 𝑘𝑆) ∈ V
130129a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑢𝐴) ∧ 𝑦 ∈ (𝐹𝑢)) → (𝑦𝑢 / 𝑘𝑆) ∈ V)
131 ovex 7464 . . . . . . . . . . . . . . . . 17 ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆) ∈ V
132 nfcv 2903 . . . . . . . . . . . . . . . . . 18 𝑘𝑢
133 nfcv 2903 . . . . . . . . . . . . . . . . . . 19 𝑘(𝐹𝑢)
134133, 53, 88nfov 7461 . . . . . . . . . . . . . . . . . 18 𝑘((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆)
135 fveq2 6907 . . . . . . . . . . . . . . . . . . 19 (𝑘 = 𝑢 → (𝐹𝑘) = (𝐹𝑢))
136135, 91oveq12d 7449 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 → ((𝐹𝑘) ↾t 𝑆) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
137132, 134, 136, 65fvmptf 7037 . . . . . . . . . . . . . . . . 17 ((𝑢𝐴 ∧ ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆) ∈ V) → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
138131, 137mpan2 691 . . . . . . . . . . . . . . . 16 (𝑢𝐴 → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
139138adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐴) → ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) = ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆))
140139eleq2d 2825 . . . . . . . . . . . . . 14 ((𝜑𝑢𝐴) → (𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↔ 𝑣 ∈ ((𝐹𝑢) ↾t 𝑢 / 𝑘𝑆)))
141 nfv 1912 . . . . . . . . . . . . . . . . 17 𝑘(𝜑𝑢𝐴)
142 nfcsb1v 3933 . . . . . . . . . . . . . . . . . 18 𝑘𝑢 / 𝑘𝑊
14388, 142nfel 2918 . . . . . . . . . . . . . . . . 17 𝑘𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊
144141, 143nfim 1894 . . . . . . . . . . . . . . . 16 𝑘((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)
145 eleq1w 2822 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢 → (𝑘𝐴𝑢𝐴))
146145anbi2d 630 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → ((𝜑𝑘𝐴) ↔ (𝜑𝑢𝐴)))
147 csbeq1a 3922 . . . . . . . . . . . . . . . . . 18 (𝑘 = 𝑢𝑊 = 𝑢 / 𝑘𝑊)
14891, 147eleq12d 2833 . . . . . . . . . . . . . . . . 17 (𝑘 = 𝑢 → (𝑆𝑊𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊))
149146, 148imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑘 = 𝑢 → (((𝜑𝑘𝐴) → 𝑆𝑊) ↔ ((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)))
150144, 149, 13chvarfv 2238 . . . . . . . . . . . . . . 15 ((𝜑𝑢𝐴) → 𝑢 / 𝑘𝑆𝑢 / 𝑘𝑊)
151 elrest 17474 . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . 13 ((𝜑𝑢𝐴) → (𝑣 ∈ ((𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))‘𝑢) ↔ ∃𝑦 ∈ (𝐹𝑢)𝑣 = (𝑦𝑢 / 𝑘𝑆)))
154 imaeq2 6076 . . . . . . . . . . . . . . 15 (𝑣 = (𝑦𝑢 / 𝑘𝑆) → ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣) = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆)))
155154eqeq2d 2746 . . . . . . . . . . . . . 14 (𝑣 = (𝑦𝑢 / 𝑘𝑆) → (𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ 𝑣) ↔ 𝑥 = ((𝑤 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) ↦ (𝑤𝑢)) “ (𝑦𝑢 / 𝑘𝑆))))
156155adantl 481 . . . . . . . . . . . . 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 5417 . . . . . . . . . . . 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 282 . . . . . . . . . . 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 3175 . . . . . . . . . 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 2806 . . . . . . . . 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 2735 . . . . . . . . . . 11 (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆)) = (𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)) ↦ (𝑥X𝑘𝐴 𝑆))
162161rnmpt 5971 . . . . . . . . . 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 3283 . . . . . . . . . . 11 𝑥𝑥 ∈ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))𝑦 = (𝑥X𝑘𝐴 𝑆)
164 nfv 1912 . . . . . . . . . . 11 𝑦𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)
16527mptex 7243 . . . . . . . . . . . . . . . 16 (𝑤 (∏t𝐹) ↦ (𝑤𝑢)) ∈ V
166165cnvex 7948 . . . . . . . . . . . . . . 15 (𝑤 (∏t𝐹) ↦ (𝑤𝑢)) ∈ V
167166imaex 7937 . . . . . . . . . . . . . 14 ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∈ V
168167rgen2w 3064 . . . . . . . . . . . . 13 𝑢𝐴𝑣 ∈ (𝐹𝑢)((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∈ V
169 ineq1 4221 . . . . . . . . . . . . . . 15 (𝑥 = ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) → (𝑥X𝑘𝐴 𝑆) = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆))
170169eqeq2d 2746 . . . . . . . . . . . . . 14 (𝑥 = ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) → (𝑦 = (𝑥X𝑘𝐴 𝑆) ↔ 𝑦 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)))
1716, 170rexrnmpo 7573 . . . . . . . . . . . . 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 2739 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑦 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ 𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)))
1741732rexbidv 3220 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑦 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆) ↔ ∃𝑢𝐴𝑣 ∈ (𝐹𝑢)𝑥 = (((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣) ∩ X𝑘𝐴 𝑆)))
175172, 174bitrid 283 . . . . . . . . . . 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 2811 . . . . . . . . . 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 2763 . . . . . . . . 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 2735 . . . . . . . . . 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 7566 . . . . . . . . 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 2800 . . . . . . . 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 4179 . . . . . . 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 2787 . . . . . 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 2775 . . . . 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 6911 . . . 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 2789 . . 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 6911 . 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 2735 . . . . . 6 (∏t𝐹) = (∏t𝐹)
18872, 187, 6ptval2 23625 . . . . 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 7446 . . 3 (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = ((topGen‘(fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣))))) ↾t X𝑘𝐴 𝑆))
191 fvex 6920 . . . 4 (fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ∈ V
192 tgrest 23183 . . . 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 2778 . 2 (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = (topGen‘((fi‘({ (∏t𝐹)} ∪ ran (𝑢𝐴, 𝑣 ∈ (𝐹𝑢) ↦ ((𝑤 (∏t𝐹) ↦ (𝑤𝑢)) “ 𝑣)))) ↾t X𝑘𝐴 𝑆)))
195 eqid 2735 . . . 4 (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆)))
19679, 195, 178ptval2 23625 . . 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 2785 1 (𝜑 → ((∏t𝐹) ↾t X𝑘𝐴 𝑆) = (∏t‘(𝑘𝐴 ↦ ((𝐹𝑘) ↾t 𝑆))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  csb 3908  cun 3961  cin 3962  wss 3963  {csn 4631  cop 4637   cuni 4912  cmpt 5231  ccnv 5688  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  Xcixp 8936  ficfi 9448  t crest 17467  topGenctg 17484  tcpt 17485  Topctop 22915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-2o 8506  df-ixp 8937  df-en 8985  df-dom 8986  df-fin 8988  df-fi 9449  df-rest 17469  df-topgen 17490  df-pt 17491  df-top 22916  df-topon 22933  df-bases 22969
This theorem is referenced by:  poimirlem30  37637
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