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Theorem finixpnum 33065
 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 3131 . . . 4 (𝑤 = ∅ → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ ∅ 𝐵 ∈ dom card))
2 ixpeq1 7879 . . . . . 6 (𝑤 = ∅ → X𝑥𝑤 𝐵 = X𝑥 ∈ ∅ 𝐵)
3 ixp0x 7896 . . . . . 6 X𝑥 ∈ ∅ 𝐵 = {∅}
42, 3syl6eq 2671 . . . . 5 (𝑤 = ∅ → X𝑥𝑤 𝐵 = {∅})
54eleq1d 2683 . . . 4 (𝑤 = ∅ → (X𝑥𝑤 𝐵 ∈ dom card ↔ {∅} ∈ dom card))
61, 5imbi12d 334 . . 3 (𝑤 = ∅ → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥 ∈ ∅ 𝐵 ∈ dom card → {∅} ∈ dom card)))
7 raleq 3131 . . . 4 (𝑤 = 𝑦 → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥𝑦 𝐵 ∈ dom card))
8 ixpeq1 7879 . . . . 5 (𝑤 = 𝑦X𝑥𝑤 𝐵 = X𝑥𝑦 𝐵)
98eleq1d 2683 . . . 4 (𝑤 = 𝑦 → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥𝑦 𝐵 ∈ dom card))
107, 9imbi12d 334 . . 3 (𝑤 = 𝑦 → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card)))
11 raleq 3131 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
12 ralunb 3778 . . . . . 6 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card))
13 vex 3193 . . . . . . . 8 𝑧 ∈ V
14 ralsnsg 4194 . . . . . . . . 9 (𝑧 ∈ V → (∀𝑥 ∈ {𝑧}𝐵 ∈ dom card ↔ [𝑧 / 𝑥]𝐵 ∈ dom card))
15 sbcel1g 3965 . . . . . . . . 9 (𝑧 ∈ V → ([𝑧 / 𝑥]𝐵 ∈ dom card ↔ 𝑧 / 𝑥𝐵 ∈ dom card))
1614, 15bitrd 268 . . . . . . . 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 729 . . . . . 6 ((∀𝑥𝑦 𝐵 ∈ dom card ∧ ∀𝑥 ∈ {𝑧}𝐵 ∈ dom card) ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card))
1912, 18bitri 264 . . . . 5 (∀𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card))
2011, 19syl6bb 276 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (∀𝑥𝑤 𝐵 ∈ dom card ↔ (∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card)))
21 ixpeq1 7879 . . . . 5 (𝑤 = (𝑦 ∪ {𝑧}) → X𝑥𝑤 𝐵 = X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
2221eleq1d 2683 . . . 4 (𝑤 = (𝑦 ∪ {𝑧}) → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
2320, 22imbi12d 334 . . 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 3131 . . . 4 (𝑤 = 𝐴 → (∀𝑥𝑤 𝐵 ∈ dom card ↔ ∀𝑥𝐴 𝐵 ∈ dom card))
25 ixpeq1 7879 . . . . 5 (𝑤 = 𝐴X𝑥𝑤 𝐵 = X𝑥𝐴 𝐵)
2625eleq1d 2683 . . . 4 (𝑤 = 𝐴 → (X𝑥𝑤 𝐵 ∈ dom card ↔ X𝑥𝐴 𝐵 ∈ dom card))
2724, 26imbi12d 334 . . 3 (𝑤 = 𝐴 → ((∀𝑥𝑤 𝐵 ∈ dom card → X𝑥𝑤 𝐵 ∈ dom card) ↔ (∀𝑥𝐴 𝐵 ∈ dom card → X𝑥𝐴 𝐵 ∈ dom card)))
28 snfi 7998 . . . 4 {∅} ∈ Fin
29 finnum 8734 . . . 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 8737 . . . . . . . . . . 11 ((X𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card) → (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ∈ dom card)
3332ancoms 469 . . . . . . . . . 10 ((𝑧 / 𝑥𝐵 ∈ dom card ∧ X𝑥𝑦 𝐵 ∈ dom card) → (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ∈ dom card)
34 xp1st 7158 . . . . . . . . . . . . . . . 16 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (1st𝑤) ∈ X𝑥𝑦 𝐵)
35 ixpfn 7874 . . . . . . . . . . . . . . . 16 ((1st𝑤) ∈ X𝑥𝑦 𝐵 → (1st𝑤) Fn 𝑦)
3634, 35syl 17 . . . . . . . . . . . . . . 15 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (1st𝑤) Fn 𝑦)
37 fvex 6168 . . . . . . . . . . . . . . . 16 (2nd𝑤) ∈ V
3813, 37fnsn 5914 . . . . . . . . . . . . . . 15 {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}
3936, 38jctir 560 . . . . . . . . . . . . . 14 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → ((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}))
40 disjsn 4223 . . . . . . . . . . . . . . 15 ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧𝑦)
4140biimpri 218 . . . . . . . . . . . . . 14 𝑧𝑦 → (𝑦 ∩ {𝑧}) = ∅)
42 fnun 5965 . . . . . . . . . . . . . 14 ((((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧}) ∧ (𝑦 ∩ {𝑧}) = ∅) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
4339, 41, 42syl2anr 495 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) Fn (𝑦 ∪ {𝑧}))
44 fvex 6168 . . . . . . . . . . . . . . . . 17 (1st𝑤) ∈ V
4544elixp 7875 . . . . . . . . . . . . . . . 16 ((1st𝑤) ∈ X𝑥𝑦 𝐵 ↔ ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵))
4634, 45sylib 208 . . . . . . . . . . . . . . 15 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵))
47 fvun1 6236 . . . . . . . . . . . . . . . . . . . . . 22 (((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥𝑦)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
4838, 47mp3an2 1409 . . . . . . . . . . . . . . . . . . . . 21 (((1st𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑥𝑦)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
4948anassrs 679 . . . . . . . . . . . . . . . . . . . 20 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = ((1st𝑤)‘𝑥))
5049eleq1d 2683 . . . . . . . . . . . . . . . . . . 19 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → ((((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵 ↔ ((1st𝑤)‘𝑥) ∈ 𝐵))
5150biimprd 238 . . . . . . . . . . . . . . . . . 18 ((((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) ∧ 𝑥𝑦) → (((1st𝑤)‘𝑥) ∈ 𝐵 → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5251ralimdva 2958 . . . . . . . . . . . . . . . . 17 (((1st𝑤) Fn 𝑦 ∧ (𝑦 ∩ {𝑧}) = ∅) → (∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5352ancoms 469 . . . . . . . . . . . . . . . 16 (((𝑦 ∩ {𝑧}) = ∅ ∧ (1st𝑤) Fn 𝑦) → (∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵 → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵))
5453impr 648 . . . . . . . . . . . . . . 15 (((𝑦 ∩ {𝑧}) = ∅ ∧ ((1st𝑤) Fn 𝑦 ∧ ∀𝑥𝑦 ((1st𝑤)‘𝑥) ∈ 𝐵)) → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
5541, 46, 54syl2an 494 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥𝑦 (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
56 vsnid 4187 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ {𝑧}
5741, 56jctir 560 . . . . . . . . . . . . . . . . . 18 𝑧𝑦 → ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧}))
58 fvun2 6237 . . . . . . . . . . . . . . . . . . 19 (((1st𝑤) Fn 𝑦 ∧ {⟨𝑧, (2nd𝑤)⟩} Fn {𝑧} ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
5938, 58mp3an2 1409 . . . . . . . . . . . . . . . . . 18 (((1st𝑤) Fn 𝑦 ∧ ((𝑦 ∩ {𝑧}) = ∅ ∧ 𝑧 ∈ {𝑧})) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
6036, 57, 59syl2anr 495 . . . . . . . . . . . . . . . . 17 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧))
61 csbfv 6200 . . . . . . . . . . . . . . . . 17 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑧)
6213, 37fvsn 6411 . . . . . . . . . . . . . . . . . 18 ({⟨𝑧, (2nd𝑤)⟩}‘𝑧) = (2nd𝑤)
6362eqcomi 2630 . . . . . . . . . . . . . . . . 17 (2nd𝑤) = ({⟨𝑧, (2nd𝑤)⟩}‘𝑧)
6460, 61, 633eqtr4g 2680 . . . . . . . . . . . . . . . 16 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) = (2nd𝑤))
65 xp2nd 7159 . . . . . . . . . . . . . . . . 17 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → (2nd𝑤) ∈ 𝑧 / 𝑥𝐵)
6665adantl 482 . . . . . . . . . . . . . . . 16 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → (2nd𝑤) ∈ 𝑧 / 𝑥𝐵)
6764, 66eqeltrd 2698 . . . . . . . . . . . . . . 15 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → 𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
68 ralsnsg 4194 . . . . . . . . . . . . . . . . 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 3961 . . . . . . . . . . . . . . . 16 ([𝑧 / 𝑥](((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
7169, 70bitri 264 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵𝑧 / 𝑥(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝑧 / 𝑥𝐵)
7267, 71sylibr 224 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥 ∈ {𝑧} (((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
73 ralun 3779 . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ∀𝑥 ∈ (𝑦 ∪ {𝑧})(((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})‘𝑥) ∈ 𝐵)
75 snex 4879 . . . . . . . . . . . . . . 15 {⟨𝑧, (2nd𝑤)⟩} ∈ V
7644, 75unex 6921 . . . . . . . . . . . . . 14 ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) ∈ V
7776elixp 7875 . . . . . . . . . . . . 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 697 . . . . . . . . . . . 12 ((¬ 𝑧𝑦𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)) → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) ∈ X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
79 eqid 2621 . . . . . . . . . . . 12 (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})) = (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))
8078, 79fmptd 6351 . . . . . . . . . . 11 𝑧𝑦 → (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)⟶X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
81 ixpfn 7874 . . . . . . . . . . . . . . . . 17 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑢 Fn (𝑦 ∪ {𝑧}))
82 ssun1 3760 . . . . . . . . . . . . . . . . 17 𝑦 ⊆ (𝑦 ∪ {𝑧})
83 fnssres 5972 . . . . . . . . . . . . . . . . 17 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑦 ⊆ (𝑦 ∪ {𝑧})) → (𝑢𝑦) Fn 𝑦)
8481, 82, 83sylancl 693 . . . . . . . . . . . . . . . 16 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑦) Fn 𝑦)
85 vex 3193 . . . . . . . . . . . . . . . . . 18 𝑢 ∈ V
8685elixp 7875 . . . . . . . . . . . . . . . . 17 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ↔ (𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵))
87 ssralv 3651 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 (𝑢𝑥) ∈ 𝐵))
8882, 87ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 (𝑢𝑥) ∈ 𝐵)
89 fvres 6174 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑦 → ((𝑢𝑦)‘𝑥) = (𝑢𝑥))
9089eleq1d 2683 . . . . . . . . . . . . . . . . . . . . 21 (𝑥𝑦 → (((𝑢𝑦)‘𝑥) ∈ 𝐵 ↔ (𝑢𝑥) ∈ 𝐵))
9190biimprd 238 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑦 → ((𝑢𝑥) ∈ 𝐵 → ((𝑢𝑦)‘𝑥) ∈ 𝐵))
9291ralimia 2946 . . . . . . . . . . . . . . . . . . 19 (∀𝑥𝑦 (𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9388, 92syl 17 . . . . . . . . . . . . . . . . . 18 (∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9493adantl 482 . . . . . . . . . . . . . . . . 17 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ ∀𝑥 ∈ (𝑦 ∪ {𝑧})(𝑢𝑥) ∈ 𝐵) → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9586, 94sylbi 207 . . . . . . . . . . . . . . . 16 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵)
9685resex 5412 . . . . . . . . . . . . . . . . 17 (𝑢𝑦) ∈ V
9796elixp 7875 . . . . . . . . . . . . . . . 16 ((𝑢𝑦) ∈ X𝑥𝑦 𝐵 ↔ ((𝑢𝑦) Fn 𝑦 ∧ ∀𝑥𝑦 ((𝑢𝑦)‘𝑥) ∈ 𝐵))
9884, 95, 97sylanbrc 697 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑦) ∈ X𝑥𝑦 𝐵)
99 ssun2 3761 . . . . . . . . . . . . . . . . . 18 {𝑧} ⊆ (𝑦 ∪ {𝑧})
10099, 56sselii 3585 . . . . . . . . . . . . . . . . 17 𝑧 ∈ (𝑦 ∪ {𝑧})
101 csbeq1 3522 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑧𝑤 / 𝑥𝐵 = 𝑧 / 𝑥𝐵)
102101fvixp 7873 . . . . . . . . . . . . . . . . 17 ((𝑢X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵𝑧 ∈ (𝑦 ∪ {𝑧})) → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
103100, 102mpan2 706 . . . . . . . . . . . . . . . 16 (𝑢X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵 → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
104 nfcv 2761 . . . . . . . . . . . . . . . . 17 𝑤𝐵
105 nfcsb1v 3535 . . . . . . . . . . . . . . . . 17 𝑥𝑤 / 𝑥𝐵
106 csbeq1a 3528 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤𝐵 = 𝑤 / 𝑥𝐵)
107104, 105, 106cbvixp 7885 . . . . . . . . . . . . . . . 16 X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 = X𝑤 ∈ (𝑦 ∪ {𝑧})𝑤 / 𝑥𝐵
108103, 107eleq2s 2716 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵)
109 opelxpi 5118 . . . . . . . . . . . . . . 15 (((𝑢𝑦) ∈ X𝑥𝑦 𝐵 ∧ (𝑢𝑧) ∈ 𝑧 / 𝑥𝐵) → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
11098, 108, 109syl2anc 692 . . . . . . . . . . . . . 14 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
111110adantl 482 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵))
112 disj3 3999 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∩ {𝑧}) = ∅ ↔ 𝑦 = (𝑦 ∖ {𝑧}))
11340, 112sylbb1 227 . . . . . . . . . . . . . . . . . 18 𝑧𝑦𝑦 = (𝑦 ∖ {𝑧}))
114 difun2 4026 . . . . . . . . . . . . . . . . . 18 ((𝑦 ∪ {𝑧}) ∖ {𝑧}) = (𝑦 ∖ {𝑧})
115113, 114syl6eqr 2673 . . . . . . . . . . . . . . . . 17 𝑧𝑦𝑦 = ((𝑦 ∪ {𝑧}) ∖ {𝑧}))
116115reseq2d 5366 . . . . . . . . . . . . . . . 16 𝑧𝑦 → (𝑢𝑦) = (𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})))
117116uneq1d 3750 . . . . . . . . . . . . . . 15 𝑧𝑦 → ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
118117adantr 481 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
119 fvex 6168 . . . . . . . . . . . . . . . . . . 19 (𝑢𝑧) ∈ V
12096, 119op1std 7138 . . . . . . . . . . . . . . . . . 18 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → (1st𝑤) = (𝑢𝑦))
12196, 119op2ndd 7139 . . . . . . . . . . . . . . . . . . . 20 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → (2nd𝑤) = (𝑢𝑧))
122121opeq2d 4384 . . . . . . . . . . . . . . . . . . 19 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ⟨𝑧, (2nd𝑤)⟩ = ⟨𝑧, (𝑢𝑧)⟩)
123122sneqd 4167 . . . . . . . . . . . . . . . . . 18 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → {⟨𝑧, (2nd𝑤)⟩} = {⟨𝑧, (𝑢𝑧)⟩})
124120, 123uneq12d 3752 . . . . . . . . . . . . . . . . 17 (𝑤 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
125 snex 4879 . . . . . . . . . . . . . . . . . 18 {⟨𝑧, (𝑢𝑧)⟩} ∈ V
12696, 125unex 6921 . . . . . . . . . . . . . . . . 17 ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}) ∈ V
127124, 79, 126fvmpt 6249 . . . . . . . . . . . . . . . 16 (⟨(𝑢𝑦), (𝑢𝑧)⟩ ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
128110, 127syl 17 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
129128adantl 482 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩) = ((𝑢𝑦) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
130 fnsnsplit 6415 . . . . . . . . . . . . . . . 16 ((𝑢 Fn (𝑦 ∪ {𝑧}) ∧ 𝑧 ∈ (𝑦 ∪ {𝑧})) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
13181, 100, 130sylancl 693 . . . . . . . . . . . . . . 15 (𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
132131adantl 482 . . . . . . . . . . . . . 14 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑢 ↾ ((𝑦 ∪ {𝑧}) ∖ {𝑧})) ∪ {⟨𝑧, (𝑢𝑧)⟩}))
133118, 129, 1323eqtr4rd 2666 . . . . . . . . . . . . 13 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → 𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩))
134 fveq2 6158 . . . . . . . . . . . . . . 15 (𝑣 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣) = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩))
135134eqeq2d 2631 . . . . . . . . . . . . . 14 (𝑣 = ⟨(𝑢𝑦), (𝑢𝑧)⟩ → (𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣) ↔ 𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘⟨(𝑢𝑦), (𝑢𝑧)⟩)))
136135rspcev 3299 . . . . . . . . . . . . 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))
137111, 133, 136syl2anc 692 . . . . . . . . . . . 12 ((¬ 𝑧𝑦𝑢X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵) → ∃𝑣 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣))
138137ralrimiva 2962 . . . . . . . . . . 11 𝑧𝑦 → ∀𝑢X 𝑥 ∈ (𝑦 ∪ {𝑧})𝐵𝑣 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)𝑢 = ((𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩}))‘𝑣))
139 dffo3 6340 . . . . . . . . . . 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)))
14080, 138, 139sylanbrc 697 . . . . . . . . . 10 𝑧𝑦 → (𝑤 ∈ (X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵) ↦ ((1st𝑤) ∪ {⟨𝑧, (2nd𝑤)⟩})):(X𝑥𝑦 𝐵 × 𝑧 / 𝑥𝐵)–ontoX𝑥 ∈ (𝑦 ∪ {𝑧})𝐵)
141 fonum 8841 . . . . . . . . . 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)
14233, 140, 141syl2anr 495 . . . . . . . . 9 ((¬ 𝑧𝑦 ∧ (𝑧 / 𝑥𝐵 ∈ dom card ∧ X𝑥𝑦 𝐵 ∈ dom card)) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)
143142expr 642 . . . . . . . 8 ((¬ 𝑧𝑦𝑧 / 𝑥𝐵 ∈ dom card) → (X𝑥𝑦 𝐵 ∈ dom card → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card))
14431, 143syl9r 78 . . . . . . 7 ((¬ 𝑧𝑦𝑧 / 𝑥𝐵 ∈ dom card) → (∀𝑥𝑦 𝐵 ∈ dom card → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
145144expimpd 628 . . . . . 6 𝑧𝑦 → ((𝑧 / 𝑥𝐵 ∈ dom card ∧ ∀𝑥𝑦 𝐵 ∈ dom card) → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
146145ancomsd 470 . . . . 5 𝑧𝑦 → ((∀𝑥𝑦 𝐵 ∈ dom card ∧ 𝑧 / 𝑥𝐵 ∈ dom card) → ((∀𝑥𝑦 𝐵 ∈ dom card → X𝑥𝑦 𝐵 ∈ dom card) → X𝑥 ∈ (𝑦 ∪ {𝑧})𝐵 ∈ dom card)))
147146com23 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)))
148147adantl 482 . . 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)))
1496, 10, 23, 27, 30, 148findcard2s 8161 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴 𝐵 ∈ dom card → X𝑥𝐴 𝐵 ∈ dom card))
150149imp 445 1 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴 𝐵 ∈ dom card) → X𝑥𝐴 𝐵 ∈ dom card)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∀wral 2908  ∃wrex 2909  Vcvv 3190  [wsbc 3422  ⦋csb 3519   ∖ cdif 3557   ∪ cun 3558   ∩ cin 3559   ⊆ wss 3560  ∅c0 3897  {csn 4155  ⟨cop 4161   ↦ cmpt 4683   × cxp 5082  dom cdm 5084   ↾ cres 5086   Fn wfn 5852  ⟶wf 5853  –onto→wfo 5855  ‘cfv 5857  1st c1st 7126  2nd c2nd 7127  Xcixp 7868  Fincfn 7915  cardccrd 8721 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525  df-er 7702  df-map 7819  df-ixp 7869  df-en 7916  df-dom 7917  df-fin 7919  df-card 8725  df-acn 8728 This theorem is referenced by:  poimirlem32  33112
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