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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  ontowfo 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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