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Theorem omxpenlem 8351
Description: Lemma for omxpen 8352. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 25-May-2015.)
Hypothesis
Ref Expression
omxpenlem.1 𝐹 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
Assertion
Ref Expression
omxpenlem ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem omxpenlem
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eloni 5988 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
21ad2antlr 717 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → Ord 𝐵)
3 simprl 761 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑥𝐵)
4 ordsucss 7298 . . . . . . . 8 (Ord 𝐵 → (𝑥𝐵 → suc 𝑥𝐵))
52, 3, 4sylc 65 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → suc 𝑥𝐵)
6 onelon 6003 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
76ad2ant2lr 738 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑥 ∈ On)
8 suceloni 7293 . . . . . . . . 9 (𝑥 ∈ On → suc 𝑥 ∈ On)
97, 8syl 17 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → suc 𝑥 ∈ On)
10 simplr 759 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝐵 ∈ On)
11 simpll 757 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝐴 ∈ On)
12 omwordi 7937 . . . . . . . 8 ((suc 𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑥𝐵 → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵)))
139, 10, 11, 12syl3anc 1439 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (suc 𝑥𝐵 → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵)))
145, 13mpd 15 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵))
15 simprr 763 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑦𝐴)
16 onelon 6003 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
1716ad2ant2rl 739 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑦 ∈ On)
18 omcl 7902 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
1911, 7, 18syl2anc 579 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·o 𝑥) ∈ On)
20 oaord 7913 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (𝑦𝐴 ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴)))
2117, 11, 19, 20syl3anc 1439 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝑦𝐴 ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴)))
2215, 21mpbid 224 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴))
23 omsuc 7892 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
2411, 7, 23syl2anc 579 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
2522, 24eleqtrrd 2862 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o suc 𝑥))
2614, 25sseldd 3822 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
2726ralrimivva 3153 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑥𝐵𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
28 omxpenlem.1 . . . . 5 𝐹 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
2928fmpt2 7519 . . . 4 (∀𝑥𝐵𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ↔ 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
3027, 29sylib 210 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
3130ffnd 6294 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐵 × 𝐴))
32 simpll 757 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ∈ On)
33 omcl 7902 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
34 onelon 6003 . . . . . . . 8 (((𝐴 ·o 𝐵) ∈ On ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
3533, 34sylan 575 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
36 noel 4146 . . . . . . . . . . . 12 ¬ 𝑚 ∈ ∅
37 oveq1 6931 . . . . . . . . . . . . . 14 (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
38 om0r 7905 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
3937, 38sylan9eqr 2836 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
4039eleq2d 2845 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝑚 ∈ (𝐴 ·o 𝐵) ↔ 𝑚 ∈ ∅))
4136, 40mtbiri 319 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ¬ 𝑚 ∈ (𝐴 ·o 𝐵))
4241ex 403 . . . . . . . . . 10 (𝐵 ∈ On → (𝐴 = ∅ → ¬ 𝑚 ∈ (𝐴 ·o 𝐵)))
4342necon2ad 2984 . . . . . . . . 9 (𝐵 ∈ On → (𝑚 ∈ (𝐴 ·o 𝐵) → 𝐴 ≠ ∅))
4443adantl 475 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑚 ∈ (𝐴 ·o 𝐵) → 𝐴 ≠ ∅))
4544imp 397 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ≠ ∅)
46 omeu 7951 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑚 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
4732, 35, 45, 46syl3anc 1439 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
48 vex 3401 . . . . . . . . 9 𝑚 ∈ V
49 vex 3401 . . . . . . . . 9 𝑛 ∈ V
5048, 49brcnv 5552 . . . . . . . 8 (𝑚𝐹𝑛𝑛𝐹𝑚)
51 eleq1 2847 . . . . . . . . . . . . . . . . 17 (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) → (𝑚 ∈ (𝐴 ·o 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)))
5251biimpac 472 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (𝐴 ·o 𝐵) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
536ex 403 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∈ On → (𝑥𝐵𝑥 ∈ On))
5453ad2antlr 717 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) → (𝑥𝐵𝑥 ∈ On))
55 simplll 765 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
56 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
5755, 56, 18syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
58 simplrr 768 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑦𝐴)
5955, 58, 16syl2anc 579 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
60 oaword1 7918 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ·o 𝑥) ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑥) ⊆ ((𝐴 ·o 𝑥) +o 𝑦))
6157, 59, 60syl2anc 579 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ⊆ ((𝐴 ·o 𝑥) +o 𝑦))
62 simplrl 767 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
6333ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
64 ontr2 6025 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ·o 𝑥) ∈ On ∧ (𝐴 ·o 𝐵) ∈ On) → (((𝐴 ·o 𝑥) ⊆ ((𝐴 ·o 𝑥) +o 𝑦) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵)))
6557, 63, 64syl2anc 579 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (((𝐴 ·o 𝑥) ⊆ ((𝐴 ·o 𝑥) +o 𝑦) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵)))
6661, 62, 65mp2and 689 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵))
67 simpllr 766 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
6862ne0d 4150 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ≠ ∅)
69 on0eln0 6033 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ·o 𝐵) ∈ On → (∅ ∈ (𝐴 ·o 𝐵) ↔ (𝐴 ·o 𝐵) ≠ ∅))
7063, 69syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (𝐴 ·o 𝐵) ≠ ∅))
7168, 70mpbird 249 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ (𝐴 ·o 𝐵))
72 om00el 7942 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
7372ad2antrr 716 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
7471, 73mpbid 224 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
7574simpld 490 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ 𝐴)
76 omord2 7933 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑥𝐵 ↔ (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵)))
7756, 67, 55, 75, 76syl31anc 1441 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝑥𝐵 ↔ (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵)))
7866, 77mpbird 249 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑥𝐵)
7978ex 403 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐵))
8054, 79impbid 204 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) → (𝑥𝐵𝑥 ∈ On))
8180expr 450 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → (𝑦𝐴 → (𝑥𝐵𝑥 ∈ On)))
8281pm5.32rd 573 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴)))
8352, 82sylan2 586 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑚 ∈ (𝐴 ·o 𝐵) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴)))
8483expr 450 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴))))
8584pm5.32rd 573 . . . . . . . . . . . . 13 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))))
86 eqcom 2785 . . . . . . . . . . . . . 14 (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)
8786anbi2i 616 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ 𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
8885, 87syl6bb 279 . . . . . . . . . . . 12 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
8988anbi2d 622 . . . . . . . . . . 11 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ((𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))) ↔ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))))
90 an12 635 . . . . . . . . . . 11 ((𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)) ↔ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
9189, 90syl6bb 279 . . . . . . . . . 10 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ((𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))) ↔ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))))
92912exbidv 1967 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))))
93 df-mpt2 6929 . . . . . . . . . . . 12 (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))}
94 dfoprab2 6980 . . . . . . . . . . . 12 {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))} = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}
9528, 93, 943eqtri 2806 . . . . . . . . . . 11 𝐹 = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}
9695breqi 4894 . . . . . . . . . 10 (𝑛𝐹𝑚𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}𝑚)
97 df-br 4889 . . . . . . . . . 10 (𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}𝑚 ↔ ⟨𝑛, 𝑚⟩ ∈ {⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))})
98 opabid 5221 . . . . . . . . . 10 (⟨𝑛, 𝑚⟩ ∈ {⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))} ↔ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))))
9996, 97, 983bitri 289 . . . . . . . . 9 (𝑛𝐹𝑚 ↔ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))))
100 r2ex 3246 . . . . . . . . 9 (∃𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚) ↔ ∃𝑥𝑦((𝑥 ∈ On ∧ 𝑦𝐴) ∧ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
10192, 99, 1003bitr4g 306 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑛𝐹𝑚 ↔ ∃𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
10250, 101syl5bb 275 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑚𝐹𝑛 ↔ ∃𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
103102eubidv 2606 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (∃!𝑛 𝑚𝐹𝑛 ↔ ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
10447, 103mpbird 249 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ∃!𝑛 𝑚𝐹𝑛)
105104ralrimiva 3148 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚𝐹𝑛)
106 fnres 6255 . . . 4 ((𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚𝐹𝑛)
107105, 106sylibr 226 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵))
108 relcnv 5759 . . . . 5 Rel 𝐹
109 df-rn 5368 . . . . . 6 ran 𝐹 = dom 𝐹
11030frnd 6300 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (𝐴 ·o 𝐵))
111109, 110syl5eqssr 3869 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → dom 𝐹 ⊆ (𝐴 ·o 𝐵))
112 relssres 5689 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ (𝐴 ·o 𝐵)) → (𝐹 ↾ (𝐴 ·o 𝐵)) = 𝐹)
113108, 111, 112sylancr 581 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ (𝐴 ·o 𝐵)) = 𝐹)
114113fneq1d 6228 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ 𝐹 Fn (𝐴 ·o 𝐵)))
115107, 114mpbid 224 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐴 ·o 𝐵))
116 dff1o4 6401 . 2 (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵) ↔ (𝐹 Fn (𝐵 × 𝐴) ∧ 𝐹 Fn (𝐴 ·o 𝐵)))
11731, 115, 116sylanbrc 578 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386   = wceq 1601  wex 1823  wcel 2107  ∃!weu 2586  wne 2969  wral 3090  wrex 3091  wss 3792  c0 4141  cop 4404   class class class wbr 4888  {copab 4950   × cxp 5355  ccnv 5356  dom cdm 5357  ran crn 5358  cres 5359  Rel wrel 5362  Ord word 5977  Oncon0 5978  suc csuc 5980   Fn wfn 6132  wf 6133  1-1-ontowf1o 6136  (class class class)co 6924  {coprab 6925  cmpt2 6926   +o coa 7842   ·o comu 7843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-omul 7850
This theorem is referenced by:  omxpen  8352  omf1o  8353  infxpenc  9176
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