MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij1lem9 Structured version   Visualization version   GIF version

Theorem ackbij1lem9 9088
Description: Lemma for ackbij1 9098. (Contributed by Stefan O'Rear, 19-Nov-2014.)
Hypothesis
Ref Expression
ackbij.f 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
Assertion
Ref Expression
ackbij1lem9 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐹‘(𝐴𝐵)) = ((𝐹𝐴) +𝑜 (𝐹𝐵)))
Distinct variable groups:   𝑥,𝐹,𝑦   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem ackbij1lem9
StepHypRef Expression
1 inss2 3867 . . . . . . . . . 10 (𝒫 ω ∩ Fin) ⊆ Fin
21sseli 3632 . . . . . . . . 9 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
323ad2ant1 1102 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ Fin)
4 snfi 8079 . . . . . . . . . 10 {𝑦} ∈ Fin
5 inss1 3866 . . . . . . . . . . . . . . . 16 (𝒫 ω ∩ Fin) ⊆ 𝒫 ω
65sseli 3632 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
76elpwid 4203 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
873ad2ant1 1102 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ ω)
9 onfin2 8193 . . . . . . . . . . . . . 14 ω = (On ∩ Fin)
10 inss2 3867 . . . . . . . . . . . . . 14 (On ∩ Fin) ⊆ Fin
119, 10eqsstri 3668 . . . . . . . . . . . . 13 ω ⊆ Fin
128, 11syl6ss 3648 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ Fin)
1312sselda 3636 . . . . . . . . . . 11 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
14 pwfi 8302 . . . . . . . . . . 11 (𝑦 ∈ Fin ↔ 𝒫 𝑦 ∈ Fin)
1513, 14sylib 208 . . . . . . . . . 10 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → 𝒫 𝑦 ∈ Fin)
16 xpfi 8272 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
174, 15, 16sylancr 696 . . . . . . . . 9 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
1817ralrimiva 2995 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19 iunfi 8295 . . . . . . . 8 ((𝐴 ∈ Fin ∧ ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
203, 18, 19syl2anc 694 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
21 ficardid 8826 . . . . . . 7 ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐴 ({𝑦} × 𝒫 𝑦))
2220, 21syl 17 . . . . . 6 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐴 ({𝑦} × 𝒫 𝑦))
231sseli 3632 . . . . . . . . 9 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
24233ad2ant2 1103 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ Fin)
255sseli 3632 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
2625elpwid 4203 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
27263ad2ant2 1103 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ ω)
2827, 11syl6ss 3648 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ Fin)
2928sselda 3636 . . . . . . . . . . 11 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → 𝑦 ∈ Fin)
3029, 14sylib 208 . . . . . . . . . 10 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → 𝒫 𝑦 ∈ Fin)
314, 30, 16sylancr 696 . . . . . . . . 9 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
3231ralrimiva 2995 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33 iunfi 8295 . . . . . . . 8 ((𝐵 ∈ Fin ∧ ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
3424, 32, 33syl2anc 694 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
35 ficardid 8826 . . . . . . 7 ( 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
3634, 35syl 17 . . . . . 6 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
37 cdaen 9033 . . . . . 6 (((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∧ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) → ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
3822, 36, 37syl2anc 694 . . . . 5 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
39 djudisj 5596 . . . . . . . 8 ((𝐴𝐵) = ∅ → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
40393ad2ant3 1104 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
41 cdaun 9032 . . . . . . 7 (( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin ∧ 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin ∧ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
4220, 34, 40, 41syl3anc 1366 . . . . . 6 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
43 iunxun 4637 . . . . . 6 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦) = ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
4442, 43syl6breqr 4727 . . . . 5 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦))
45 entr 8049 . . . . 5 ((((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ∧ ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) +𝑐 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)) → ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦))
4638, 44, 45syl2anc 694 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦))
47 carden2b 8831 . . . 4 (((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))) ≈ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦) → (card‘((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)))
4846, 47syl 17 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))) = (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)))
49 ficardom 8825 . . . . 5 ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
5020, 49syl 17 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω)
51 ficardom 8825 . . . . 5 ( 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin → (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
5234, 51syl 17 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω)
53 nnacda 9061 . . . 4 (((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) ∈ ω ∧ (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) ∈ ω) → (card‘((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑜 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
5450, 52, 53syl2anc 694 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑐 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑜 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
5548, 54eqtr3d 2687 . 2 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑜 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
56 ackbij1lem6 9085 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))
57563adant3 1101 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))
58 ackbij.f . . . 4 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘ 𝑦𝑥 ({𝑦} × 𝒫 𝑦)))
5958ackbij1lem7 9086 . . 3 ((𝐴𝐵) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴𝐵)) = (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)))
6057, 59syl 17 . 2 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐹‘(𝐴𝐵)) = (card‘ 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦)))
6158ackbij1lem7 9086 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
6258ackbij1lem7 9086 . . . 4 (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐵) = (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
6361, 62oveqan12d 6709 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝐴) +𝑜 (𝐹𝐵)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑜 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
64633adant3 1101 . 2 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ((𝐹𝐴) +𝑜 (𝐹𝐵)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑜 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
6555, 60, 643eqtr4d 2695 1 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → (𝐹‘(𝐴𝐵)) = ((𝐹𝐴) +𝑜 (𝐹𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  cun 3605  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210   ciun 4552   class class class wbr 4685  cmpt 4762   × cxp 5141  Oncon0 5761  cfv 5926  (class class class)co 6690  ωcom 7107   +𝑜 coa 7602  cen 7994  Fincfn 7997  cardccrd 8799   +𝑐 ccda 9027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028
This theorem is referenced by:  ackbij1lem12  9091  ackbij1lem13  9092  ackbij1lem14  9093  ackbij1lem16  9095  ackbij1lem18  9097
  Copyright terms: Public domain W3C validator