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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  ontowfo 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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