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Theorem omxpenlem 9112
Description: Lemma for omxpen 9113. (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 6396 . . . . . . . . 9 (𝐵 ∈ On → Ord 𝐵)
21ad2antlr 727 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → Ord 𝐵)
3 simprl 771 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑥𝐵)
4 ordsucss 7838 . . . . . . . 8 (Ord 𝐵 → (𝑥𝐵 → suc 𝑥𝐵))
52, 3, 4sylc 65 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → suc 𝑥𝐵)
6 onelon 6411 . . . . . . . . . 10 ((𝐵 ∈ On ∧ 𝑥𝐵) → 𝑥 ∈ On)
76ad2ant2lr 748 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑥 ∈ On)
8 onsuc 7831 . . . . . . . . 9 (𝑥 ∈ On → suc 𝑥 ∈ On)
97, 8syl 17 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → suc 𝑥 ∈ On)
10 simplr 769 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝐵 ∈ On)
11 simpll 767 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝐴 ∈ On)
12 omwordi 8608 . . . . . . . 8 ((suc 𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑥𝐵 → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵)))
139, 10, 11, 12syl3anc 1370 . . . . . . 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 773 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑦𝐴)
16 onelon 6411 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑦𝐴) → 𝑦 ∈ On)
1716ad2ant2rl 749 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → 𝑦 ∈ On)
18 omcl 8573 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
1911, 7, 18syl2anc 584 . . . . . . . . 9 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·o 𝑥) ∈ On)
20 oaord 8584 . . . . . . . . 9 ((𝑦 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·o 𝑥) ∈ On) → (𝑦𝐴 ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴)))
2117, 11, 19, 20syl3anc 1370 . . . . . . . 8 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝑦𝐴 ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴)))
2215, 21mpbid 232 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴))
23 omsuc 8563 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
2411, 7, 23syl2anc 584 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
2522, 24eleqtrrd 2842 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o suc 𝑥))
2614, 25sseldd 3996 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥𝐵𝑦𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
2726ralrimivva 3200 . . . 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 8092 . . . 4 (∀𝑥𝐵𝑦𝐴 ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ↔ 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
3027, 29sylib 218 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
3130ffnd 6738 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐵 × 𝐴))
32 simpll 767 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ∈ On)
33 omcl 8573 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
34 onelon 6411 . . . . . . . 8 (((𝐴 ·o 𝐵) ∈ On ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
3533, 34sylan 580 . . . . . . 7 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
36 noel 4344 . . . . . . . . . . . 12 ¬ 𝑚 ∈ ∅
37 oveq1 7438 . . . . . . . . . . . . . 14 (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
38 om0r 8576 . . . . . . . . . . . . . 14 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
3937, 38sylan9eqr 2797 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
4039eleq2d 2825 . . . . . . . . . . . 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 2953 . . . . . . . . 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 8622 . . . . . . 7 ((𝐴 ∈ On ∧ 𝑚 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
4732, 35, 45, 46syl3anc 1370 . . . . . 6 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ∃!𝑛𝑥 ∈ On ∃𝑦𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
48 vex 3482 . . . . . . . . 9 𝑚 ∈ V
49 vex 3482 . . . . . . . . 9 𝑛 ∈ V
5048, 49brcnv 5896 . . . . . . . 8 (𝑚𝐹𝑛𝑛𝐹𝑚)
51 eleq1 2827 . . . . . . . . . . . . . . . . 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 727 . . . . . . . . . . . . . . . . . . 19 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) → (𝑥𝐵𝑥 ∈ On))
55 simplll 775 . . . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
58 simplrr 778 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑦𝐴)
5955, 58, 16syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
60 oaword1 8589 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐴 ·o 𝑥) ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑥) ⊆ ((𝐴 ·o 𝑥) +o 𝑦))
6157, 59, 60syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
64 ontr2 6433 . . . . . . . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵))
67 simpllr 776 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
6862ne0d 4348 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ≠ ∅)
69 on0eln0 6442 . . . . . . . . . . . . . . . . . . . . . . . . . 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 8613 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
7372ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
7471, 73mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 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 8604 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑥𝐵 ↔ (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵)))
7756, 67, 55, 75, 76syl31anc 1372 . . . . . . . . . . . . . . . . . . . . 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 212 . . . . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . . . . 16 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → ((𝑥𝐵𝑦𝐴) ↔ (𝑥 ∈ On ∧ 𝑦𝐴)))
8352, 82sylan2 593 . . . . . . . . . . . . . . 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 578 . . . . . . . . . . . . 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 2742 . . . . . . . . . . . . . 14 (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)
8786anbi2i 623 . . . . . . . . . . . . 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 630 . . . . . . . . . . 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 645 . . . . . . . . . . 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 1922 . . . . . . . . 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 7436 . . . . . . . . . . . 12 (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))}
94 dfoprab2 7491 . . . . . . . . . . . 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 2767 . . . . . . . . . . 11 𝐹 = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}
9695breqi 5154 . . . . . . . . . 10 (𝑛𝐹𝑚𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐵𝑦𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}𝑚)
97 df-br 5149 . . . . . . . . . 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 5534 . . . . . . . . . 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 3194 . . . . . . . . 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 2584 . . . . . 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 3144 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚𝐹𝑛)
106 fnres 6696 . . . 4 ((𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚𝐹𝑛)
107105, 106sylibr 234 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵))
108 relcnv 6125 . . . . 5 Rel 𝐹
109 df-rn 5700 . . . . . 6 ran 𝐹 = dom 𝐹
11030frnd 6745 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (𝐴 ·o 𝐵))
111109, 110eqsstrrid 4045 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → dom 𝐹 ⊆ (𝐴 ·o 𝐵))
112 relssres 6042 . . . . 5 ((Rel 𝐹 ∧ dom 𝐹 ⊆ (𝐴 ·o 𝐵)) → (𝐹 ↾ (𝐴 ·o 𝐵)) = 𝐹)
113108, 111, 112sylancr 587 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹 ↾ (𝐴 ·o 𝐵)) = 𝐹)
114113fneq1d 6662 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ 𝐹 Fn (𝐴 ·o 𝐵)))
115107, 114mpbid 232 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐴 ·o 𝐵))
116 dff1o4 6857 . 2 (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵) ↔ (𝐹 Fn (𝐵 × 𝐴) ∧ 𝐹 Fn (𝐴 ·o 𝐵)))
11731, 115, 116sylanbrc 583 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 206  wa 395   = wceq 1537  wex 1776  wcel 2106  ∃!weu 2566  wne 2938  wral 3059  wrex 3068  wss 3963  c0 4339  cop 4637   class class class wbr 5148  {copab 5210   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  Rel wrel 5694  Ord word 6385  Oncon0 6386  suc csuc 6388   Fn wfn 6558  wf 6559  1-1-ontowf1o 6562  (class class class)co 7431  {coprab 7432  cmpo 7433   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510
This theorem is referenced by:  omxpen  9113  omf1o  9114  infxpenc  10056
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