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Theorem omxpenlem 8012
 Description: Lemma for omxpen 8013. (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 5697 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
21ad2antlr 762 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → Ord 𝐵)
3 simprl 793 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑥𝐵)
4 ordsucss 6972 . . . . . . . 8 (Ord 𝐵 → (𝑥𝐵 → suc 𝑥𝐵))
52, 3, 4sylc 65 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → suc 𝑥𝐵)
6 onelon 5712 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
76ad2ant2lr 783 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑥 ∈ On)
8 suceloni 6967 . . . . . . . . 9 (𝑥 ∈ On → suc 𝑥 ∈ On)
97, 8syl 17 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → suc 𝑥 ∈ On)
10 simplr 791 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝐵 ∈ On)
11 simpll 789 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝐴 ∈ On)
12 omwordi 7603 . . . . . . . 8 ((suc 𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑥𝐵 → (𝐴 ·𝑜 suc 𝑥) ⊆ (𝐴 ·𝑜 𝐵)))
139, 10, 11, 12syl3anc 1323 . . . . . . 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 795 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑦𝐴)
16 onelon 5712 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
1716ad2ant2rl 784 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑦 ∈ On)
18 omcl 7568 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
1911, 7, 18syl2anc 692 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·𝑜 𝑥) ∈ On)
20 oaord 7579 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (𝑦𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
2117, 11, 19, 20syl3anc 1323 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝑦𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
2215, 21mpbid 222 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
23 omsuc 7558 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
2411, 7, 23syl2anc 692 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
2522, 24eleqtrrd 2701 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 suc 𝑥))
2614, 25sseldd 3588 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
2726ralrimivva 2966 . . . 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 7189 . . . 4 (∀𝑥𝐵𝑦𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ↔ 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵))
3027, 29sylib 208 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵))
31 ffn 6007 . . 3 (𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵) → 𝐹 Fn (𝐵 × 𝐴))
3230, 31syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐵 × 𝐴))
33 simpll 789 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝐴 ∈ On)
34 omcl 7568 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
35 onelon 5712 . . . . . . . 8 (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝑚 ∈ On)
3634, 35sylan 488 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝑚 ∈ On)
37 noel 3900 . . . . . . . . . . . 12 ¬ 𝑚 ∈ ∅
38 oveq1 6617 . . . . . . . . . . . . . 14 (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
39 om0r 7571 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
4038, 39sylan9eqr 2677 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
4140eleq2d 2684 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) ↔ 𝑚 ∈ ∅))
4237, 41mtbiri 317 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ¬ 𝑚 ∈ (𝐴 ·𝑜 𝐵))
4342ex 450 . . . . . . . . . 10 (𝐵 ∈ On → (𝐴 = ∅ → ¬ 𝑚 ∈ (𝐴 ·𝑜 𝐵)))
4443necon2ad 2805 . . . . . . . . 9 (𝐵 ∈ On → (𝑚 ∈ (𝐴 ·𝑜 𝐵) → 𝐴 ≠ ∅))
4544adantl 482 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) → 𝐴 ≠ ∅))
4645imp 445 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝐴 ≠ ∅)
47 omeu 7617 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑚 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))
4833, 36, 46, 47syl3anc 1323 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))
49 vex 3192 . . . . . . . . 9 𝑚 ∈ V
50 vex 3192 . . . . . . . . 9 𝑛 ∈ V
5149, 50brcnv 5270 . . . . . . . 8 (𝑚𝐹𝑛𝑛𝐹𝑚)
52 eleq1 2686 . . . . . . . . . . . . . . . . 17 (𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)))
5352biimpac 503 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
546ex 450 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∈ On → (𝑥𝐵𝑥 ∈ On))
5554ad2antlr 762 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) → (𝑥𝐵𝑥 ∈ On))
56 simplll 797 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
57 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
5856, 57, 18syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
59 simplrr 800 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑦𝐴)
6056, 59, 16syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
61 oaword1 7584 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑥) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
6258, 60, 61syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
63 simplrl 799 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
6434ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
65 ontr2 5736 . . . . . . . . . . . . . . . . . . . . . . 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)))
6658, 64, 65syl2anc 692 . . . . . . . . . . . . . . . . . . . . . 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)))
6762, 63, 66mp2and 714 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ (𝐴 ·𝑜 𝐵))
68 simpllr 798 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
69 ne0i 3902 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (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) ≠ ∅)
71 on0eln0 5744 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐴 ·𝑜 𝐵) ∈ On → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (𝐴 ·𝑜 𝐵) ≠ ∅))
7264, 71syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (𝐴 ·𝑜 𝐵) ≠ ∅))
7370, 72mpbird 247 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ (𝐴 ·𝑜 𝐵))
74 om00el 7608 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
7574ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
7673, 75mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
7776simpld 475 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ 𝐴)
78 omord2 7599 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑥𝐵 ↔ (𝐴 ·𝑜 𝑥) ∈ (𝐴 ·𝑜 𝐵)))
7957, 68, 56, 77, 78syl31anc 1326 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝑥𝐵 ↔ (𝐴 ·𝑜 𝑥) ∈ (𝐴 ·𝑜 𝐵)))
8067, 79mpbird 247 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑥𝐵)
8180ex 450 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐵))
8255, 81impbid 202 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦𝐴)) → (𝑥𝐵𝑥 ∈ On))
8382expr 642 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)) → (𝑦𝐴 → (𝑥𝐵𝑥 ∈ On)))
8483pm5.32rd 671 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴)))
8553, 84sylan2 491 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑚 ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴)))
8685expr 642 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴))))
8786pm5.32rd 671 . . . . . . . . . . . . 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))))
88 eqcom 2628 . . . . . . . . . . . . . 14 (𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)
8988anbi2i 729 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ 𝑦𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))
9087, 89syl6bb 276 . . . . . . . . . . . 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)))
9190anbi2d 739 . . . . . . . . . . 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))))
92 an12 837 . . . . . . . . . . 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)))
9391, 92syl6bb 276 . . . . . . . . . 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))))
94932exbidv 1849 . . . . . . . . 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))))
95 df-mpt2 6615 . . . . . . . . . . . 12 (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))}
96 dfoprab2 6661 . . . . . . . . . . . 12 {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))} = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}
9728, 95, 963eqtri 2647 . . . . . . . . . . 11 𝐹 = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}
9897breqi 4624 . . . . . . . . . 10 (𝑛𝐹𝑚𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}𝑚)
99 df-br 4619 . . . . . . . . . 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)))})
100 opabid 4947 . . . . . . . . . 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))))
10198, 99, 1003bitri 286 . . . . . . . . 9 (𝑛𝐹𝑚 ↔ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))))
102 r2ex 3055 . . . . . . . . 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)))
10394, 101, 1023bitr4g 303 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (𝑛𝐹𝑚 ↔ ∃𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
10451, 103syl5bb 272 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (𝑚𝐹𝑛 ↔ ∃𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
105104eubidv 2489 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (∃!𝑛 𝑚𝐹𝑛 ↔ ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
10648, 105mpbird 247 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → ∃!𝑛 𝑚𝐹𝑛)
107106ralrimiva 2961 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑚 ∈ (𝐴 ·𝑜 𝐵)∃!𝑛 𝑚𝐹𝑛)
108 fnres 5970 . . . 4 ((𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵) ↔ ∀𝑚 ∈ (𝐴 ·𝑜 𝐵)∃!𝑛 𝑚𝐹𝑛)
109107, 108sylibr 224 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵))
110 relcnv 5467 . . . . 5 Rel 𝐹
111 df-rn 5090 . . . . . 6 ran 𝐹 = dom 𝐹
112 frn 6015 . . . . . . 7 (𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵) → ran 𝐹 ⊆ (𝐴 ·𝑜 𝐵))
11330, 112syl 17 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (𝐴 ·𝑜 𝐵))
114111, 113syl5eqssr 3634 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → dom 𝐹 ⊆ (𝐴 ·𝑜 𝐵))
115 relssres 5401 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ (𝐴 ·𝑜 𝐵)) → (𝐹 ↾ (𝐴 ·𝑜 𝐵)) = 𝐹)
116110, 114, 115sylancr 694 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ (𝐴 ·𝑜 𝐵)) = 𝐹)
117116fneq1d 5944 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵) ↔ 𝐹 Fn (𝐴 ·𝑜 𝐵)))
118109, 117mpbid 222 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐴 ·𝑜 𝐵))
119 dff1o4 6107 . 2 (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵) ↔ (𝐹 Fn (𝐵 × 𝐴) ∧ 𝐹 Fn (𝐴 ·𝑜 𝐵)))
12032, 118, 119sylanbrc 697 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 196   ∧ wa 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987  ∃!weu 2469   ≠ wne 2790  ∀wral 2907  ∃wrex 2908   ⊆ wss 3559  ∅c0 3896  ⟨cop 4159   class class class wbr 4618  {copab 4677   × cxp 5077  ◡ccnv 5078  dom cdm 5079  ran crn 5080   ↾ cres 5081  Rel wrel 5084  Ord word 5686  Oncon0 5687  suc csuc 5689   Fn wfn 5847  ⟶wf 5848  –1-1-onto→wf1o 5851  (class class class)co 6610  {coprab 6611   ↦ cmpt2 6612   +𝑜 coa 7509   ·𝑜 comu 7510 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 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 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 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-omul 7517 This theorem is referenced by:  omxpen  8013  omf1o  8014  infxpenc  8792
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