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

Theorem ackbij1lem9 8994
Description: Lemma for ackbij1 9004. (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 3812 . . . . . . . . . 10 (𝒫 ω ∩ Fin) ⊆ Fin
21sseli 3579 . . . . . . . . 9 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
323ad2ant1 1080 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ Fin)
4 snfi 7982 . . . . . . . . . 10 {𝑦} ∈ Fin
5 inss1 3811 . . . . . . . . . . . . . . . 16 (𝒫 ω ∩ Fin) ⊆ 𝒫 ω
65sseli 3579 . . . . . . . . . . . . . . 15 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
76elpwid 4141 . . . . . . . . . . . . . 14 (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
873ad2ant1 1080 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ ω)
9 onfin2 8096 . . . . . . . . . . . . . 14 ω = (On ∩ Fin)
10 inss2 3812 . . . . . . . . . . . . . 14 (On ∩ Fin) ⊆ Fin
119, 10eqsstri 3614 . . . . . . . . . . . . 13 ω ⊆ Fin
128, 11syl6ss 3595 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ Fin)
1312sselda 3583 . . . . . . . . . . 11 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → 𝑦 ∈ Fin)
14 pwfi 8205 . . . . . . . . . . 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 8175 . . . . . . . . . 10 (({𝑦} ∈ Fin ∧ 𝒫 𝑦 ∈ Fin) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
174, 15, 16sylancr 694 . . . . . . . . 9 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐴) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
1817ralrimiva 2960 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
19 iunfi 8198 . . . . . . . 8 ((𝐴 ∈ Fin ∧ ∀𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin) → 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
203, 18, 19syl2anc 692 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∈ Fin)
21 ficardid 8732 . . . . . . 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 3579 . . . . . . . . 9 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
24233ad2ant2 1081 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ Fin)
255sseli 3579 . . . . . . . . . . . . . . 15 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
2625elpwid 4141 . . . . . . . . . . . . . 14 (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
27263ad2ant2 1081 . . . . . . . . . . . . 13 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ ω)
2827, 11syl6ss 3595 . . . . . . . . . . . 12 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ Fin)
2928sselda 3583 . . . . . . . . . . 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 694 . . . . . . . . 9 (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) ∧ 𝑦𝐵) → ({𝑦} × 𝒫 𝑦) ∈ Fin)
3231ralrimiva 2960 . . . . . . . 8 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
33 iunfi 8198 . . . . . . . 8 ((𝐵 ∈ Fin ∧ ∀𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin) → 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
3424, 32, 33syl2anc 692 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → 𝑦𝐵 ({𝑦} × 𝒫 𝑦) ∈ Fin)
35 ficardid 8732 . . . . . . 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 8939 . . . . . 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 692 . . . . 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 5520 . . . . . . . 8 ((𝐴𝐵) = ∅ → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
40393ad2ant3 1082 . . . . . . 7 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴𝐵) = ∅) → ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∩ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)) = ∅)
41 cdaun 8938 . . . . . . 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 1323 . . . . . 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 4571 . . . . . 6 𝑦 ∈ (𝐴𝐵)({𝑦} × 𝒫 𝑦) = ( 𝑦𝐴 ({𝑦} × 𝒫 𝑦) ∪ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))
4442, 43syl6breqr 4655 . . . . 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 7952 . . . . 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 692 . . . 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 8737 . . . 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 8731 . . . . 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 8731 . . . . 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 8967 . . . 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 692 . . 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 2657 . 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 8991 . . . 4 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴𝐵) ∈ (𝒫 ω ∩ Fin))
57563adant3 1079 . . 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 8992 . . 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 8992 . . . 4 (𝐴 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐴) = (card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)))
6258ackbij1lem7 8992 . . . 4 (𝐵 ∈ (𝒫 ω ∩ Fin) → (𝐹𝐵) = (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦)))
6361, 62oveqan12d 6623 . . 3 ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹𝐴) +𝑜 (𝐹𝐵)) = ((card‘ 𝑦𝐴 ({𝑦} × 𝒫 𝑦)) +𝑜 (card‘ 𝑦𝐵 ({𝑦} × 𝒫 𝑦))))
64633adant3 1079 . 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 2665 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cun 3553  cin 3554  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148   ciun 4485   class class class wbr 4613  cmpt 4673   × cxp 5072  Oncon0 5682  cfv 5847  (class class class)co 6604  ωcom 7012   +𝑜 coa 7502  cen 7896  Fincfn 7899  cardccrd 8705   +𝑐 ccda 8933
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 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  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 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-cda 8934
This theorem is referenced by:  ackbij1lem12  8997  ackbij1lem13  8998  ackbij1lem14  8999  ackbij1lem16  9001  ackbij1lem18  9003
  Copyright terms: Public domain W3C validator