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Theorem finixpnum 37612
Description: A finite Cartesian product of numerable sets is numerable. (Contributed by Brendan Leahy, 24-Feb-2019.)
Assertion
Ref Expression
finixpnum ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ dom card) → X𝑥𝐴 𝐵 ∈ dom card)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem finixpnum
Dummy variables 𝑣 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raleq 3323 . . . 4 (𝑤 = ∅ → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ dom card))
2 ixpeq1 8948 . . . . . 6 (𝑤 = ∅ → X𝑥𝑤 𝐵 = X𝑥 ∈ ∅ 𝐵)
3 ixp0x 8966 . . . . . 6 X𝑥 ∈ ∅ 𝐵 = {∅}
42, 3eqtrdi 2793 . . . . 5 (𝑤 = ∅ → X𝑥𝑤 𝐵 = {∅})
54eleq1d 2826 . . . 4 (𝑤 = ∅ → (X𝑥𝑤 𝐵 ∈ dom card ↔ {∅} ∈ dom card))
61, 5imbi12d 344 . . 3 (𝑤 = ∅ → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)))
7 raleq 3323 . . . 4 (𝑤 = 𝑦 → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥𝑦 𝐵 ∈ dom card))
8 ixpeq1 8948 . . . . 5 (𝑤 = 𝑦X𝑥𝑤 𝐵 = X𝑥𝑦 𝐵)
98eleq1d 2826 . . . 4 (𝑤 = 𝑦 → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥𝑦 𝐵 ∈ dom card))
107, 9imbi12d 344 . . 3 (𝑤 = 𝑦 → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card)))
11 raleq 3323 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
12 ralunb 4197 . . . . . 6 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card))
13 vex 3484 . . . . . . . 8 𝑧 ∈ V
14 ralsnsg 4670 . . . . . . . . 9 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ [𝑧 / 𝑥]𝐵 ∈ dom card))
15 sbcel1g 4416 . . . . . . . . 9 (𝑧 ∈ V → ([𝑧 / 𝑥]𝐵 ∈ dom card ↔ 𝑧 / 𝑥𝐵 ∈ dom card))
1614, 15bitrd 279 . . . . . . . 8 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ 𝑧 / 𝑥𝐵 ∈ dom card))
1713, 16ax-mp 5 . . . . . . 7 (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ 𝑧 / 𝑥𝐵 ∈ dom card)
1817anbi2i 623 . . . . . 6 ((∀𝑥𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card) ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card))
1912, 18bitri 275 . . . . 5 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card))
2011, 19bitrdi 287 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card)))
21 ixpeq1 8948 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → X𝑥𝑤 𝐵 = X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
2221eleq1d 2826 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
2320, 22imbi12d 344 . . 3 (𝑤 = (𝑦 ∪ {𝑧}) → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ ((∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
24 raleq 3323 . . . 4 (𝑤 = 𝐴 → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥𝐴 𝐵 ∈ dom card))
25 ixpeq1 8948 . . . . 5 (𝑤 = 𝐴X𝑥𝑤 𝐵 = X𝑥𝐴 𝐵)
2625eleq1d 2826 . . . 4 (𝑤 = 𝐴 → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥𝐴 𝐵 ∈ dom card))
2724, 26imbi12d 344 . . 3 (𝑤 = 𝐴 → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥𝐴 𝐵 ∈ dom card → X𝑥𝐴 𝐵 ∈ dom card)))
28 snfi 9083 . . . 4 {∅} ∈ Fin
29 finnum 9988 . . . 4 ({∅} ∈ Fin → {∅} ∈ dom card)
3028, 29mp1i 13 . . 3 (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)
31 pm2.27 42 . . . . . . . 8 (∀𝑥𝑦 𝐵 ∈ dom card → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → X𝑥𝑦 𝐵 ∈ dom card))
32 xpnum 9991 . . . . . . . . . . 11 ((X𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card) → (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ∈ dom card)
3332ancoms 458 . . . . . . . . . 10 ((𝑧 / 𝑥𝐵 ∈ dom card ∧ X𝑥𝑦 𝐵 ∈ dom card) → (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ∈ dom card)
34 xp1st 8046 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (1st𝑤) ∈ X𝑥𝑦 𝐵)
35 ixpfn 8943 . . . . . . . . . . . . . . . 16 ((1st𝑤) ∈ X𝑥𝑦 𝐵 → (1st𝑤) Fn 𝑦)
3634, 35syl 17 . . . . . . . . . . . . . . 15 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (1st𝑤) Fn 𝑦)
37 fvex 6919 . . . . . . . . . . . . . . . 16 (2nd𝑤) ∈ V
3813, 37fnsn 6624 . . . . . . . . . . . . . . 15 {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}
3936, 38jctir 520 . . . . . . . . . . . . . 14 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → ((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}))
40 disjsn 4711 . . . . . . . . . . . . . . 15 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
4140biimpri 228 . . . . . . . . . . . . . 14 𝑧𝑦 → (𝑦 ∩ {𝑧}) = ∅)
42 fnun 6682 . . . . . . . . . . . . . 14 ((((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
4339, 41, 42syl2anr 597 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
44 fvex 6919 . . . . . . . . . . . . . . . . 17 (1st𝑤) ∈ V
4544elixp 8944 . . . . . . . . . . . . . . . 16 ((1st𝑤) ∈ X𝑥𝑦 𝐵 ↔ ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵))
4634, 45sylib 218 . . . . . . . . . . . . . . 15 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵))
47 fvun1 7000 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥𝑦)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
4838, 47mp3an2 1451 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥𝑦)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
4948anassrs 467 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
5049eleq1d 2826 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → ((((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ((1st𝑤)‘𝑥) ∈ 𝐵))
5150biimprd 248 . . . . . . . . . . . . . . . . . 18 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → (((1st𝑤)‘𝑥) ∈ 𝐵 → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5251ralimdva 3167 . . . . . . . . . . . . . . . . 17 (((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) → (∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5352ancoms 458 . . . . . . . . . . . . . . . 16 (((𝑦 ∩ {𝑧}) = ∅ ∧ (1st𝑤) Fn 𝑦) → (∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5453impr 454 . . . . . . . . . . . . . . 15 (((𝑦 ∩ {𝑧}) = ∅ ∧ ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵)) → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
5541, 46, 54syl2an 596 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
56 vsnid 4663 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ {𝑧}
5741, 56jctir 520 . . . . . . . . . . . . . . . . . 18 𝑧𝑦 → ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧}))
58 fvun2 7001 . . . . . . . . . . . . . . . . . . 19 (((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
5938, 58mp3an2 1451 . . . . . . . . . . . . . . . . . 18 (((1st𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
6036, 57, 59syl2anr 597 . . . . . . . . . . . . . . . . 17 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
61 csbfv 6956 . . . . . . . . . . . . . . . . 17 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧)
6213, 37fvsn 7201 . . . . . . . . . . . . . . . . . 18 ({⟨𝑧, (2nd𝑤)⟩}‘𝑧) = (2nd𝑤)
6362eqcomi 2746 . . . . . . . . . . . . . . . . 17 (2nd𝑤) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧)
6460, 61, 633eqtr4g 2802 . . . . . . . . . . . . . . . 16 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = (2nd𝑤))
65 xp2nd 8047 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (2nd𝑤) ∈ 𝑧 / 𝑥𝐵)
6665adantl 481 . . . . . . . . . . . . . . . 16 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → (2nd𝑤) ∈ 𝑧 / 𝑥𝐵)
6764, 66eqeltrd 2841 . . . . . . . . . . . . . . 15 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
68 ralsnsg 4670 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵[𝑧 / 𝑥](((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
6913, 68ax-mp 5 . . . . . . . . . . . . . . . 16 (∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵[𝑧 / 𝑥](((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
70 sbcel12 4411 . . . . . . . . . . . . . . . 16 ([𝑧 / 𝑥](((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
7169, 70bitri 275 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
7267, 71sylibr 234 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
73 ralun 4198 . . . . . . . . . . . . . 14 ((∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵 ∧ ∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
7455, 72, 73syl2anc 584 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
75 snex 5436 . . . . . . . . . . . . . . 15 {⟨𝑧, (2nd𝑤)⟩} ∈ V
7644, 75unex 7764 . . . . . . . . . . . . . 14 ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) ∈ V
7776elixp 8944 . . . . . . . . . . . . 13 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
7843, 74, 77sylanbrc 583 . . . . . . . . . . . 12 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
7978fmpttd 7135 . . . . . . . . . . 11 𝑧𝑦 → (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)⟶X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
80 ixpfn 8943 . . . . . . . . . . . . . . . . 17 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑢 Fn (𝑦 ∪ {𝑧}))
81 ssun1 4178 . . . . . . . . . . . . . . . . 17 𝑦 ⊆ (𝑦 ∪ {𝑧})
82 fnssres 6691 . . . . . . . . . . . . . . . . 17 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑢𝑦) Fn 𝑦)
8380, 81, 82sylancl 586 . . . . . . . . . . . . . . . 16 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑦) Fn 𝑦)
84 vex 3484 . . . . . . . . . . . . . . . . . 18 𝑢 ∈ V
8584elixp 8944 . . . . . . . . . . . . . . . . 17 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵))
86 ssralv 4052 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 (𝑢𝑥) ∈ 𝐵))
8781, 86ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 (𝑢𝑥) ∈ 𝐵)
88 fvres 6925 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑦 → ((𝑢𝑦)‘𝑥) = (𝑢𝑥))
8988eleq1d 2826 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑦 → (((𝑢𝑦)‘𝑥) ∈ 𝐵 ↔ (𝑢𝑥) ∈ 𝐵))
9089biimprd 248 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑦 → ((𝑢𝑥) ∈ 𝐵 → ((𝑢𝑦)‘𝑥) ∈ 𝐵))
9190ralimia 3080 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑦 (𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9287, 91syl 17 . . . . . . . . . . . . . . . . . 18 (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9392adantl 481 . . . . . . . . . . . . . . . . 17 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵) → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9485, 93sylbi 217 . . . . . . . . . . . . . . . 16 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9584resex 6047 . . . . . . . . . . . . . . . . 17 (𝑢𝑦) ∈ V
9695elixp 8944 . . . . . . . . . . . . . . . 16 ((𝑢𝑦) ∈ X𝑥𝑦 𝐵 ↔ ((𝑢𝑦) Fn 𝑦 ∧ ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵))
9783, 94, 96sylanbrc 583 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑦) ∈ X𝑥𝑦 𝐵)
98 ssun2 4179 . . . . . . . . . . . . . . . . . 18 {𝑧} ⊆ (𝑦 ∪ {𝑧})
9998, 56sselii 3980 . . . . . . . . . . . . . . . . 17 𝑧 ∈ (𝑦 ∪ {𝑧})
100 csbeq1 3902 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
101100fvixp 8942 . . . . . . . . . . . . . . . . 17 ((𝑢X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
10299, 101mpan2 691 . . . . . . . . . . . . . . . 16 (𝑢X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵 → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
103 nfcv 2905 . . . . . . . . . . . . . . . . 17 𝑤𝐵
104 nfcsb1v 3923 . . . . . . . . . . . . . . . . 17 𝑥𝑤 / 𝑥𝐵
105 csbeq1a 3913 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
106103, 104, 105cbvixp 8954 . . . . . . . . . . . . . . . 16 X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵
107102, 106eleq2s 2859 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
108 opelxpi 5722 . . . . . . . . . . . . . . 15 (((𝑢𝑦) ∈ X𝑥𝑦 𝐵 ∧ (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵) → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
10997, 107, 108syl2anc 584 . . . . . . . . . . . . . 14 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
110109adantl 481 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
111 disj3 4454 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∩ {𝑧}) = ∅ ↔ 𝑦 = (𝑦 ∖ {𝑧}))
11240, 111sylbb1 237 . . . . . . . . . . . . . . . . . 18 𝑧𝑦𝑦 = (𝑦 ∖ {𝑧}))
113 difun2 4481 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
114112, 113eqtr4di 2795 . . . . . . . . . . . . . . . . 17 𝑧𝑦𝑦 = ((𝑦 ∪ {𝑧}) ∖ {𝑧}))
115114reseq2d 5997 . . . . . . . . . . . . . . . 16 𝑧𝑦 → (𝑢𝑦) = (𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})))
116115uneq1d 4167 . . . . . . . . . . . . . . 15 𝑧𝑦 → ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
117116adantr 480 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
118 fvex 6919 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑧) ∈ V
11995, 118op1std 8024 . . . . . . . . . . . . . . . . . 18 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → (1st𝑤) = (𝑢𝑦))
12095, 118op2ndd 8025 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → (2nd𝑤) = (𝑢𝑧))
121120opeq2d 4880 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ⟨𝑧, (2nd𝑤)⟩ = ⟨𝑧, (𝑢𝑧)⟩)
122121sneqd 4638 . . . . . . . . . . . . . . . . . 18 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → {⟨𝑧, (2nd𝑤)⟩} = {⟨𝑧, (𝑢𝑧)⟩})
123119, 122uneq12d 4169 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
124 eqid 2737 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})) = (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))
125 snex 5436 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, (𝑢𝑧)⟩} ∈ V
12695, 125unex 7764 . . . . . . . . . . . . . . . . 17 ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) ∈ V
127123, 124, 126fvmpt 7016 . . . . . . . . . . . . . . . 16 (⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
128109, 127syl 17 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
129128adantl 481 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
130 fnsnsplit 7204 . . . . . . . . . . . . . . . 16 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
13180, 99, 130sylancl 586 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
132131adantl 481 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
133117, 129, 1323eqtr4rd 2788 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩))
134 fveq2 6906 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣) = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩))
135134rspceeqv 3645 . . . . . . . . . . . . 13 ((⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ∧ 𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩)) → ∃𝑣 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣))
136110, 133, 135syl2anc 584 . . . . . . . . . . . 12 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ∃𝑣 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣))
137136ralrimiva 3146 . . . . . . . . . . 11 𝑧𝑦 → ∀𝑢X 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑣 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣))
138 dffo3 7122 . . . . . . . . . . 11 ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)–ontoX𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)⟶X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∧ ∀𝑢X 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑣 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣)))
13979, 137, 138sylanbrc 583 . . . . . . . . . 10 𝑧𝑦 → (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)–ontoX𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
140 fonum 10098 . . . . . . . . . 10 (((X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ∈ dom card ∧ (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)–ontoX𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
14133, 139, 140syl2anr 597 . . . . . . . . 9 ((¬ 𝑧𝑦 ∧ (𝑧 / 𝑥𝐵 ∈ dom card ∧ X𝑥𝑦 𝐵 ∈ dom card)) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
142141expr 456 . . . . . . . 8 ((¬ 𝑧𝑦𝑧 / 𝑥𝐵 ∈ dom card) → (X𝑥𝑦 𝐵 ∈ dom card → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
14331, 142syl9r 78 . . . . . . 7 ((¬ 𝑧𝑦𝑧 / 𝑥𝐵 ∈ dom card) → (∀𝑥𝑦 𝐵 ∈ dom card → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
144143expimpd 453 . . . . . 6 𝑧𝑦 → ((𝑧 / 𝑥𝐵 ∈ dom card ∧ ∀𝑥𝑦 𝐵 ∈ dom card) → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
145144ancomsd 465 . . . . 5 𝑧𝑦 → ((∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card) → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
146145com23 86 . . . 4 𝑧𝑦 → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → ((∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
147146adantl 481 . . 3 ((𝑦 ∈ Fin ∧ ¬ 𝑧𝑦) → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → ((∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
1486, 10, 23, 27, 30, 147findcard2s 9205 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴 𝐵 ∈ dom card → X𝑥𝐴 𝐵 ∈ dom card))
149148imp 406 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ dom card) → X𝑥𝐴 𝐵 ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  Vcvv 3480  [wsbc 3788  csb 3899  cdif 3948  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cop 4632  cmpt 5225   × cxp 5683  dom cdm 5685  cres 5687   Fn wfn 6556  wf 6557  ontowfo 6559  cfv 6561  1st c1st 8012  2nd c2nd 8013  Xcixp 8937  Fincfn 8985  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-fin 8989  df-card 9979  df-acn 9982
This theorem is referenced by:  poimirlem32  37659
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