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Theorem omxpenlem 9068
Description: Lemma for omxpen 9069. (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 6364 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
21ad2antlr 724 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → Ord 𝐵)
3 simprl 768 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑥𝐵)
4 ordsucss 7799 . . . . . . . 8 (Ord 𝐵 → (𝑥𝐵 → suc 𝑥𝐵))
52, 3, 4sylc 65 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → suc 𝑥𝐵)
6 onelon 6379 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
76ad2ant2lr 745 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑥 ∈ On)
8 onsuc 7792 . . . . . . . . 9 (𝑥 ∈ On → suc 𝑥 ∈ On)
97, 8syl 17 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → suc 𝑥 ∈ On)
10 simplr 766 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝐵 ∈ On)
11 simpll 764 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝐴 ∈ On)
12 omwordi 8566 . . . . . . . 8 ((suc 𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑥𝐵 → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵)))
139, 10, 11, 12syl3anc 1368 . . . . . . 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 770 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑦𝐴)
16 onelon 6379 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
1716ad2ant2rl 746 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑦 ∈ On)
18 omcl 8531 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
1911, 7, 18syl2anc 583 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·o 𝑥) ∈ On)
20 oaord 8542 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (𝑦𝐴 ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴)))
2117, 11, 19, 20syl3anc 1368 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝑦𝐴 ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴)))
2215, 21mpbid 231 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴))
23 omsuc 8521 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
2411, 7, 23syl2anc 583 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
2522, 24eleqtrrd 2828 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o suc 𝑥))
2614, 25sseldd 3975 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
2726ralrimivva 3192 . . . 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))
2928fmpo 8047 . . . 4 (∀𝑥𝐵𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ↔ 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
3027, 29sylib 217 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
3130ffnd 6708 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐵 × 𝐴))
32 simpll 764 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ∈ On)
33 omcl 8531 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
34 onelon 6379 . . . . . . . 8 (((𝐴 ·o 𝐵) ∈ On ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
3533, 34sylan 579 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
36 noel 4322 . . . . . . . . . . . 12 ¬ 𝑚 ∈ ∅
37 oveq1 7408 . . . . . . . . . . . . . 14 (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
38 om0r 8534 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
3937, 38sylan9eqr 2786 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
4039eleq2d 2811 . . . . . . . . . . . 12 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝑚 ∈ (𝐴 ·o 𝐵) ↔ 𝑚 ∈ ∅))
4136, 40mtbiri 327 . . . . . . . . . . 11 ((𝐵 ∈ On ∧ 𝐴 = ∅) → ¬ 𝑚 ∈ (𝐴 ·o 𝐵))
4241ex 412 . . . . . . . . . 10 (𝐵 ∈ On → (𝐴 = ∅ → ¬ 𝑚 ∈ (𝐴 ·o 𝐵)))
4342necon2ad 2947 . . . . . . . . 9 (𝐵 ∈ On → (𝑚 ∈ (𝐴 ·o 𝐵) → 𝐴 ≠ ∅))
4443adantl 481 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑚 ∈ (𝐴 ·o 𝐵) → 𝐴 ≠ ∅))
4544imp 406 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ≠ ∅)
46 omeu 8580 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑚 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
4732, 35, 45, 46syl3anc 1368 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
48 vex 3470 . . . . . . . . 9 𝑚 ∈ V
49 vex 3470 . . . . . . . . 9 𝑛 ∈ V
5048, 49brcnv 5872 . . . . . . . 8 (𝑚𝐹𝑛𝑛𝐹𝑚)
51 eleq1 2813 . . . . . . . . . . . . . . . . 17 (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) → (𝑚 ∈ (𝐴 ·o 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)))
5251biimpac 478 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (𝐴 ·o 𝐵) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
536ex 412 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∈ On → (𝑥𝐵𝑥 ∈ On))
5453ad2antlr 724 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) → (𝑥𝐵𝑥 ∈ On))
55 simplll 772 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
56 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
5755, 56, 18syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
58 simplrr 775 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑦𝐴)
5955, 58, 16syl2anc 583 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
60 oaword1 8547 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ·o 𝑥) ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑥) ⊆ ((𝐴 ·o 𝑥) +o 𝑦))
6157, 59, 60syl2anc 583 . . . . . . . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . . . . 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 723 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
64 ontr2 6401 . . . . . . . . . . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵))
67 simpllr 773 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
6862ne0d 4327 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ≠ ∅)
69 on0eln0 6410 . . . . . . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ (𝐴 ·o 𝐵))
72 om00el 8571 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
7372ad2antrr 723 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
7471, 73mpbid 231 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
7574simpld 494 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ 𝐴)
76 omord2 8562 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑥𝐵 ↔ (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵)))
7756, 67, 55, 75, 76syl31anc 1370 . . . . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . . . . . 20 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑥𝐵)
7978ex 412 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) → (𝑥 ∈ On → 𝑥𝐵))
8054, 79impbid 211 . . . . . . . . . . . . . . . . . 18 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) → (𝑥𝐵𝑥 ∈ On))
8180expr 456 . . . . . . . . . . . . . . . . 17 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → (𝑦𝐴 → (𝑥𝐵𝑥 ∈ On)))
8281pm5.32rd 577 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴)))
8352, 82sylan2 592 . . . . . . . . . . . . . . 15 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑚 ∈ (𝐴 ·o 𝐵) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴)))
8483expr 456 . . . . . . . . . . . . . 14 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴))))
8584pm5.32rd 577 . . . . . . . . . . . . 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 2731 . . . . . . . . . . . . . 14 (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)
8786anbi2i 622 . . . . . . . . . . . . 13 (((𝑥 ∈ On ∧ 𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦𝐴) ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
8885, 87bitrdi 287 . . . . . . . . . . . 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 628 . . . . . . . . . . 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 642 . . . . . . . . . . 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, 90bitrdi 287 . . . . . . . . . 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 1919 . . . . . . . . 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-mpo 7406 . . . . . . . . . . . 12 (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))}
94 dfoprab2 7459 . . . . . . . . . . . 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 2756 . . . . . . . . . . 11 𝐹 = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}
9695breqi 5144 . . . . . . . . . 10 (𝑛𝐹𝑚𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}𝑚)
97 df-br 5139 . . . . . . . . . 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 opabidw 5514 . . . . . . . . . 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 297 . . . . . . . . 9 (𝑛𝐹𝑚 ↔ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))))
100 r2ex 3187 . . . . . . . . 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 314 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑛𝐹𝑚 ↔ ∃𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
10250, 101bitrid 283 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑚𝐹𝑛 ↔ ∃𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
103102eubidv 2572 . . . . . 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 257 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ∃!𝑛 𝑚𝐹𝑛)
105104ralrimiva 3138 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚𝐹𝑛)
106 fnres 6667 . . . 4 ((𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚𝐹𝑛)
107105, 106sylibr 233 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵))
108 relcnv 6093 . . . . 5 Rel 𝐹
109 df-rn 5677 . . . . . 6 ran 𝐹 = dom 𝐹
11030frnd 6715 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (𝐴 ·o 𝐵))
111109, 110eqsstrrid 4023 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → dom 𝐹 ⊆ (𝐴 ·o 𝐵))
112 relssres 6012 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ (𝐴 ·o 𝐵)) → (𝐹 ↾ (𝐴 ·o 𝐵)) = 𝐹)
113108, 111, 112sylancr 586 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ (𝐴 ·o 𝐵)) = 𝐹)
114113fneq1d 6632 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ 𝐹 Fn (𝐴 ·o 𝐵)))
115107, 114mpbid 231 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐴 ·o 𝐵))
116 dff1o4 6831 . 2 (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵) ↔ (𝐹 Fn (𝐵 × 𝐴) ∧ 𝐹 Fn (𝐴 ·o 𝐵)))
11731, 115, 116sylanbrc 582 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 205  wa 395   = wceq 1533  wex 1773  wcel 2098  ∃!weu 2554  wne 2932  wral 3053  wrex 3062  wss 3940  c0 4314  cop 4626   class class class wbr 5138  {copab 5200   × cxp 5664  ccnv 5665  dom cdm 5666  ran crn 5667  cres 5668  Rel wrel 5671  Ord word 6353  Oncon0 6354  suc csuc 6356   Fn wfn 6528  wf 6529  1-1-ontowf1o 6532  (class class class)co 7401  {coprab 7402  cmpo 7403   +o coa 8458   ·o comu 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-omul 8466
This theorem is referenced by:  omxpen  9069  omf1o  9070  infxpenc  10008
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