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Theorem finixpnum 35000
 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 3386 . . . 4 (𝑤 = ∅ → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ dom card))
2 ixpeq1 8459 . . . . . 6 (𝑤 = ∅ → X𝑥𝑤 𝐵 = X𝑥 ∈ ∅ 𝐵)
3 ixp0x 8477 . . . . . 6 X𝑥 ∈ ∅ 𝐵 = {∅}
42, 3syl6eq 2873 . . . . 5 (𝑤 = ∅ → X𝑥𝑤 𝐵 = {∅})
54eleq1d 2898 . . . 4 (𝑤 = ∅ → (X𝑥𝑤 𝐵 ∈ dom card ↔ {∅} ∈ dom card))
61, 5imbi12d 348 . . 3 (𝑤 = ∅ → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)))
7 raleq 3386 . . . 4 (𝑤 = 𝑦 → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥𝑦 𝐵 ∈ dom card))
8 ixpeq1 8459 . . . . 5 (𝑤 = 𝑦X𝑥𝑤 𝐵 = X𝑥𝑦 𝐵)
98eleq1d 2898 . . . 4 (𝑤 = 𝑦 → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥𝑦 𝐵 ∈ dom card))
107, 9imbi12d 348 . . 3 (𝑤 = 𝑦 → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card)))
11 raleq 3386 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
12 ralunb 4142 . . . . . 6 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card))
13 vex 3472 . . . . . . . 8 𝑧 ∈ V
14 ralsnsg 4582 . . . . . . . . 9 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ [𝑧 / 𝑥]𝐵 ∈ dom card))
15 sbcel1g 4337 . . . . . . . . 9 (𝑧 ∈ V → ([𝑧 / 𝑥]𝐵 ∈ dom card ↔ 𝑧 / 𝑥𝐵 ∈ dom card))
1614, 15bitrd 282 . . . . . . . 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 625 . . . . . 6 ((∀𝑥𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card) ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card))
1912, 18bitri 278 . . . . 5 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card))
2011, 19syl6bb 290 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card)))
21 ixpeq1 8459 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → X𝑥𝑤 𝐵 = X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
2221eleq1d 2898 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
2320, 22imbi12d 348 . . 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 3386 . . . 4 (𝑤 = 𝐴 → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥𝐴 𝐵 ∈ dom card))
25 ixpeq1 8459 . . . . 5 (𝑤 = 𝐴X𝑥𝑤 𝐵 = X𝑥𝐴 𝐵)
2625eleq1d 2898 . . . 4 (𝑤 = 𝐴 → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥𝐴 𝐵 ∈ dom card))
2724, 26imbi12d 348 . . 3 (𝑤 = 𝐴 → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥𝐴 𝐵 ∈ dom card → X𝑥𝐴 𝐵 ∈ dom card)))
28 snfi 8581 . . . 4 {∅} ∈ Fin
29 finnum 9365 . . . 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 9368 . . . . . . . . . . 11 ((X𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card) → (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ∈ dom card)
3332ancoms 462 . . . . . . . . . 10 ((𝑧 / 𝑥𝐵 ∈ dom card ∧ X𝑥𝑦 𝐵 ∈ dom card) → (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ∈ dom card)
34 xp1st 7707 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (1st𝑤) ∈ X𝑥𝑦 𝐵)
35 ixpfn 8454 . . . . . . . . . . . . . . . 16 ((1st𝑤) ∈ X𝑥𝑦 𝐵 → (1st𝑤) Fn 𝑦)
3634, 35syl 17 . . . . . . . . . . . . . . 15 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (1st𝑤) Fn 𝑦)
37 fvex 6665 . . . . . . . . . . . . . . . 16 (2nd𝑤) ∈ V
3813, 37fnsn 6391 . . . . . . . . . . . . . . 15 {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}
3936, 38jctir 524 . . . . . . . . . . . . . 14 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → ((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}))
40 disjsn 4621 . . . . . . . . . . . . . . 15 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
4140biimpri 231 . . . . . . . . . . . . . 14 𝑧𝑦 → (𝑦 ∩ {𝑧}) = ∅)
42 fnun 6443 . . . . . . . . . . . . . 14 ((((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
4339, 41, 42syl2anr 599 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
44 fvex 6665 . . . . . . . . . . . . . . . . 17 (1st𝑤) ∈ V
4544elixp 8455 . . . . . . . . . . . . . . . 16 ((1st𝑤) ∈ X𝑥𝑦 𝐵 ↔ ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵))
4634, 45sylib 221 . . . . . . . . . . . . . . 15 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵))
47 fvun1 6736 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥𝑦)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
4838, 47mp3an2 1446 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥𝑦)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
4948anassrs 471 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
5049eleq1d 2898 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → ((((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ((1st𝑤)‘𝑥) ∈ 𝐵))
5150biimprd 251 . . . . . . . . . . . . . . . . . 18 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → (((1st𝑤)‘𝑥) ∈ 𝐵 → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5251ralimdva 3169 . . . . . . . . . . . . . . . . 17 (((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) → (∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5352ancoms 462 . . . . . . . . . . . . . . . 16 (((𝑦 ∩ {𝑧}) = ∅ ∧ (1st𝑤) Fn 𝑦) → (∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5453impr 458 . . . . . . . . . . . . . . 15 (((𝑦 ∩ {𝑧}) = ∅ ∧ ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵)) → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
5541, 46, 54syl2an 598 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
56 vsnid 4576 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ {𝑧}
5741, 56jctir 524 . . . . . . . . . . . . . . . . . 18 𝑧𝑦 → ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧}))
58 fvun2 6737 . . . . . . . . . . . . . . . . . . 19 (((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
5938, 58mp3an2 1446 . . . . . . . . . . . . . . . . . 18 (((1st𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
6036, 57, 59syl2anr 599 . . . . . . . . . . . . . . . . 17 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
61 csbfv 6697 . . . . . . . . . . . . . . . . 17 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧)
6213, 37fvsn 6925 . . . . . . . . . . . . . . . . . 18 ({⟨𝑧, (2nd𝑤)⟩}‘𝑧) = (2nd𝑤)
6362eqcomi 2831 . . . . . . . . . . . . . . . . 17 (2nd𝑤) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧)
6460, 61, 633eqtr4g 2882 . . . . . . . . . . . . . . . 16 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = (2nd𝑤))
65 xp2nd 7708 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (2nd𝑤) ∈ 𝑧 / 𝑥𝐵)
6665adantl 485 . . . . . . . . . . . . . . . 16 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → (2nd𝑤) ∈ 𝑧 / 𝑥𝐵)
6764, 66eqeltrd 2914 . . . . . . . . . . . . . . 15 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
68 ralsnsg 4582 . . . . . . . . . . . . . . . . 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 4332 . . . . . . . . . . . . . . . 16 ([𝑧 / 𝑥](((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
7169, 70bitri 278 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
7267, 71sylibr 237 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
73 ralun 4143 . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
75 snex 5309 . . . . . . . . . . . . . . 15 {⟨𝑧, (2nd𝑤)⟩} ∈ V
7644, 75unex 7454 . . . . . . . . . . . . . 14 ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) ∈ V
7776elixp 8455 . . . . . . . . . . . . 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 586 . . . . . . . . . . . 12 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
7978fmpttd 6861 . . . . . . . . . . 11 𝑧𝑦 → (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)⟶X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
80 ixpfn 8454 . . . . . . . . . . . . . . . . 17 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑢 Fn (𝑦 ∪ {𝑧}))
81 ssun1 4123 . . . . . . . . . . . . . . . . 17 𝑦 ⊆ (𝑦 ∪ {𝑧})
82 fnssres 6450 . . . . . . . . . . . . . . . . 17 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑢𝑦) Fn 𝑦)
8380, 81, 82sylancl 589 . . . . . . . . . . . . . . . 16 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑦) Fn 𝑦)
84 vex 3472 . . . . . . . . . . . . . . . . . 18 𝑢 ∈ V
8584elixp 8455 . . . . . . . . . . . . . . . . 17 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵))
86 ssralv 4008 . . . . . . . . . . . . . . . . . . . 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 6671 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑦 → ((𝑢𝑦)‘𝑥) = (𝑢𝑥))
8988eleq1d 2898 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑦 → (((𝑢𝑦)‘𝑥) ∈ 𝐵 ↔ (𝑢𝑥) ∈ 𝐵))
9089biimprd 251 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑦 → ((𝑢𝑥) ∈ 𝐵 → ((𝑢𝑦)‘𝑥) ∈ 𝐵))
9190ralimia 3150 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑦 (𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9287, 91syl 17 . . . . . . . . . . . . . . . . . 18 (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9392adantl 485 . . . . . . . . . . . . . . . . 17 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵) → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9485, 93sylbi 220 . . . . . . . . . . . . . . . 16 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9584resex 5877 . . . . . . . . . . . . . . . . 17 (𝑢𝑦) ∈ V
9695elixp 8455 . . . . . . . . . . . . . . . 16 ((𝑢𝑦) ∈ X𝑥𝑦 𝐵 ↔ ((𝑢𝑦) Fn 𝑦 ∧ ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵))
9783, 94, 96sylanbrc 586 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑦) ∈ X𝑥𝑦 𝐵)
98 ssun2 4124 . . . . . . . . . . . . . . . . . 18 {𝑧} ⊆ (𝑦 ∪ {𝑧})
9998, 56sselii 3939 . . . . . . . . . . . . . . . . 17 𝑧 ∈ (𝑦 ∪ {𝑧})
100 csbeq1 3858 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
101100fvixp 8453 . . . . . . . . . . . . . . . . 17 ((𝑢X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
10299, 101mpan2 690 . . . . . . . . . . . . . . . 16 (𝑢X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵 → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
103 nfcv 2979 . . . . . . . . . . . . . . . . 17 𝑤𝐵
104 nfcsb1v 3879 . . . . . . . . . . . . . . . . 17 𝑥𝑤 / 𝑥𝐵
105 csbeq1a 3869 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
106103, 104, 105cbvixp 8465 . . . . . . . . . . . . . . . 16 X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵
107102, 106eleq2s 2932 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
108 opelxpi 5569 . . . . . . . . . . . . . . 15 (((𝑢𝑦) ∈ X𝑥𝑦 𝐵 ∧ (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵) → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
10997, 107, 108syl2anc 587 . . . . . . . . . . . . . 14 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
110109adantl 485 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
111 disj3 4375 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∩ {𝑧}) = ∅ ↔ 𝑦 = (𝑦 ∖ {𝑧}))
11240, 111sylbb1 240 . . . . . . . . . . . . . . . . . 18 𝑧𝑦𝑦 = (𝑦 ∖ {𝑧}))
113 difun2 4401 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
114112, 113eqtr4di 2875 . . . . . . . . . . . . . . . . 17 𝑧𝑦𝑦 = ((𝑦 ∪ {𝑧}) ∖ {𝑧}))
115114reseq2d 5831 . . . . . . . . . . . . . . . 16 𝑧𝑦 → (𝑢𝑦) = (𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})))
116115uneq1d 4113 . . . . . . . . . . . . . . 15 𝑧𝑦 → ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
117116adantr 484 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
118 fvex 6665 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑧) ∈ V
11995, 118op1std 7685 . . . . . . . . . . . . . . . . . 18 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → (1st𝑤) = (𝑢𝑦))
12095, 118op2ndd 7686 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → (2nd𝑤) = (𝑢𝑧))
121120opeq2d 4785 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ⟨𝑧, (2nd𝑤)⟩ = ⟨𝑧, (𝑢𝑧)⟩)
122121sneqd 4551 . . . . . . . . . . . . . . . . . 18 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → {⟨𝑧, (2nd𝑤)⟩} = {⟨𝑧, (𝑢𝑧)⟩})
123119, 122uneq12d 4115 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
124 eqid 2822 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})) = (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))
125 snex 5309 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, (𝑢𝑧)⟩} ∈ V
12695, 125unex 7454 . . . . . . . . . . . . . . . . 17 ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) ∈ V
127123, 124, 126fvmpt 6750 . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
130 fnsnsplit 6928 . . . . . . . . . . . . . . . 16 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
13180, 99, 130sylancl 589 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
132131adantl 485 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
133117, 129, 1323eqtr4rd 2868 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩))
134 fveq2 6652 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣) = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩))
135134rspceeqv 3613 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 12 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ∃𝑣 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣))
137136ralrimiva 3174 . . . . . . . . . . 11 𝑧𝑦 → ∀𝑢X 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑣 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣))
138 dffo3 6850 . . . . . . . . . . 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 586 . . . . . . . . . 10 𝑧𝑦 → (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)–ontoX𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
140 fonum 9473 . . . . . . . . . 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 599 . . . . . . . . 9 ((¬ 𝑧𝑦 ∧ (𝑧 / 𝑥𝐵 ∈ dom card ∧ X𝑥𝑦 𝐵 ∈ dom card)) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
142141expr 460 . . . . . . . 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 457 . . . . . 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 469 . . . . 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 485 . . 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 8747 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴 𝐵 ∈ dom card → X𝑥𝐴 𝐵 ∈ dom card))
149148imp 410 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ dom card) → X𝑥𝐴 𝐵 ∈ dom card)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∀wral 3130  ∃wrex 3131  Vcvv 3469  [wsbc 3747  ⦋csb 3855   ∖ cdif 3905   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908  ∅c0 4265  {csn 4539  ⟨cop 4545   ↦ cmpt 5122   × cxp 5530  dom cdm 5532   ↾ cres 5534   Fn wfn 6329  ⟶wf 6330  –onto→wfo 6332  ‘cfv 6334  1st c1st 7673  2nd c2nd 7674  Xcixp 8448  Fincfn 8496  cardccrd 9352 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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-se 5492  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-isom 6343  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-omul 8094  df-er 8276  df-map 8395  df-ixp 8449  df-en 8497  df-dom 8498  df-fin 8500  df-card 9356  df-acn 9359 This theorem is referenced by:  poimirlem32  35047
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