Theorem omxpenlem 7946

Description: Lemma for omxpen 7947. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 25-May-2015.)

Hypothesis

Ref	Expression
omxpenlem.1	⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((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.

Step	Hyp	Ref	Expression
1		eloni 5650	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Ord 𝐵)
2	1	ad2antlr 759	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → Ord 𝐵)
3		simprl 790	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐵)
4		ordsucss 6910	. . . . . . . 8 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → suc 𝑥 ⊆ 𝐵))
5	2, 3, 4	sylc 63	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → suc 𝑥 ⊆ 𝐵)
6		onelon 5665	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
7	6	ad2ant2lr 780	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ On)
8		suceloni 6905	. . . . . . . . 9 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
9	7, 8	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → suc 𝑥 ∈ On)
10		simplr 788	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝐵 ∈ On)
11		simpll 786	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝐴 ∈ On)
12		omwordi 7538	. . . . . . . 8 ⊢ ((suc 𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑥 ⊆ 𝐵 → (𝐴 ·𝑜 suc 𝑥) ⊆ (𝐴 ·𝑜 𝐵)))
13	9, 10, 11, 12	syl3anc 1318	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (suc 𝑥 ⊆ 𝐵 → (𝐴 ·𝑜 suc 𝑥) ⊆ (𝐴 ·𝑜 𝐵)))
14	5, 13	mpd 15	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·𝑜 suc 𝑥) ⊆ (𝐴 ·𝑜 𝐵))
15		simprr 792	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
16		onelon 5665	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
17	16	ad2ant2rl 781	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ On)
18		omcl 7503	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
19	11, 7, 18	syl2anc 691	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·𝑜 𝑥) ∈ On)
20		oaord 7514	. . . . . . . . 9 ⊢ ((𝑦 ∈ On ∧ 𝐴 ∈ On ∧ (𝐴 ·𝑜 𝑥) ∈ On) → (𝑦 ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
21	17, 11, 19, 20	syl3anc 1318	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝑦 ∈ 𝐴 ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴)))
22	15, 21	mpbid 221	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
23		omsuc 7493	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
24	11, 7, 23	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
25	22, 24	eleqtrrd 2691	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 suc 𝑥))
26	14, 25	sseldd 3569	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
27	26	ralrimivva 2954	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
28		omxpenlem.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
29	28	fmpt2 7126	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ↔ 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵))
30	27, 29	sylib 207	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵))
31		ffn 5958	. . 3 ⊢ (𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵) → 𝐹 Fn (𝐵 × 𝐴))
32	30, 31	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐵 × 𝐴))
33		simpll 786	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝐴 ∈ On)
34		omcl 7503	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
35		onelon 5665	. . . . . . . 8 ⊢ (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝑚 ∈ On)
36	34, 35	sylan 487	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝑚 ∈ On)
37		noel 3878	. . . . . . . . . . . 12 ⊢ ¬ 𝑚 ∈ ∅
38		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
39		om0r 7506	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
40	38, 39	sylan9eqr 2666	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
41	40	eleq2d 2673	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) ↔ 𝑚 ∈ ∅))
42	37, 41	mtbiri 316	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ¬ 𝑚 ∈ (𝐴 ·𝑜 𝐵))
43	42	ex 449	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (𝐴 = ∅ → ¬ 𝑚 ∈ (𝐴 ·𝑜 𝐵)))
44	43	necon2ad 2797	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝑚 ∈ (𝐴 ·𝑜 𝐵) → 𝐴 ≠ ∅))
45	44	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) → 𝐴 ≠ ∅))
46	45	imp 444	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝐴 ≠ ∅)
47		omeu 7552	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑚 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑛∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))
48	33, 36, 46, 47	syl3anc 1318	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → ∃!𝑛∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))
49		vex 3176	. . . . . . . . 9 ⊢ 𝑚 ∈ V
50		vex 3176	. . . . . . . . 9 ⊢ 𝑛 ∈ V
51	49, 50	brcnv 5227	. . . . . . . 8 ⊢ (𝑚◡𝐹𝑛 ↔ 𝑛𝐹𝑚)
52		eleq1 2676	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)))
53	52	biimpac 502	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
54	6	ex 449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∈ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
55	54	ad2antlr 759	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
56		simplll 794	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
57		simpr 476	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
58	56, 57, 18	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
59		simplrr 797	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴)
60	56, 59, 16	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
61		oaword1 7519	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑥) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
62	58, 60, 61	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
63		simplrl 796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
64	34	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
65		ontr2 5689	. . . . . . . . . . . . . . . . . . . . . . 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)))
66	58, 64, 65	syl2anc 691	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (((𝐴 ·𝑜 𝑥) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)) → (𝐴 ·𝑜 𝑥) ∈ (𝐴 ·𝑜 𝐵)))
67	62, 63, 66	mp2and 711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ (𝐴 ·𝑜 𝐵))
68		simpllr 795	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
69		ne0i 3880	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) → (𝐴 ·𝑜 𝐵) ≠ ∅)
70	63, 69	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝐵) ≠ ∅)
71		on0eln0 5697	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ·𝑜 𝐵) ∈ On → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (𝐴 ·𝑜 𝐵) ≠ ∅))
72	64, 71	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (𝐴 ·𝑜 𝐵) ≠ ∅))
73	70, 72	mpbird 246	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ (𝐴 ·𝑜 𝐵))
74		om00el 7543	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
75	74	ad2antrr 758	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
76	73, 75	mpbid 221	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
77	76	simpld 474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ 𝐴)
78		omord2 7534	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ (𝐴 ·𝑜 𝑥) ∈ (𝐴 ·𝑜 𝐵)))
79	57, 68, 56, 77, 78	syl31anc 1321	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝑥 ∈ 𝐵 ↔ (𝐴 ·𝑜 𝑥) ∈ (𝐴 ·𝑜 𝐵)))
80	67, 79	mpbird 246	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ 𝐵)
81	80	ex 449	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐵))
82	55, 81	impbid 201	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ On))
83	82	expr 641	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)) → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ On)))
84	83	pm5.32rd 670	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ 𝐴)))
85	53, 84	sylan2 490	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑚 ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ 𝐴)))
86	85	expr 641	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ 𝐴))))
87	86	pm5.32rd 670	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))))
88		eqcom 2617	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)
89	88	anbi2i 726	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))
90	87, 89	syl6bb 275	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
91	90	anbi2d 736	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → ((𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))) ↔ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))))
92		an12 834	. . . . . . . . . . 11 ⊢ ((𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
93	91, 92	syl6bb 275	. . . . . . . . . 10 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → ((𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))))
94	93	2exbidv 1839	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))) ↔ ∃𝑥∃𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚))))
95		df-mpt2 6554	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))}
96		dfoprab2 6599	. . . . . . . . . . . 12 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))} = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}
97	28, 95, 96	3eqtri 2636	. . . . . . . . . . 11 ⊢ 𝐹 = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}
98	97	breqi 4589	. . . . . . . . . 10 ⊢ (𝑛𝐹𝑚 ↔ 𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}𝑚)
99		df-br 4584	. . . . . . . . . 10 ⊢ (𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}𝑚 ↔ ⟨𝑛, 𝑚⟩ ∈ {⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))})
100		opabid 4907	. . . . . . . . . 10 ⊢ (⟨𝑛, 𝑚⟩ ∈ {⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))} ↔ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))))
101	98, 99, 100	3bitri 285	. . . . . . . . 9 ⊢ (𝑛𝐹𝑚 ↔ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))))
102		r2ex 3043	. . . . . . . . 9 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚) ↔ ∃𝑥∃𝑦((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
103	94, 101, 102	3bitr4g 302	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (𝑛𝐹𝑚 ↔ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
104	51, 103	syl5bb 271	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (𝑚◡𝐹𝑛 ↔ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
105	104	eubidv 2478	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → (∃!𝑛 𝑚◡𝐹𝑛 ↔ ∃!𝑛∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)))
106	48, 105	mpbird 246	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → ∃!𝑛 𝑚◡𝐹𝑛)
107	106	ralrimiva 2949	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑚 ∈ (𝐴 ·𝑜 𝐵)∃!𝑛 𝑚◡𝐹𝑛)
108		fnres 5921	. . . 4 ⊢ ((◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵) ↔ ∀𝑚 ∈ (𝐴 ·𝑜 𝐵)∃!𝑛 𝑚◡𝐹𝑛)
109	107, 108	sylibr 223	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵))
110		relcnv 5422	. . . . 5 ⊢ Rel ◡𝐹
111		df-rn 5049	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
112		frn 5966	. . . . . . 7 ⊢ (𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵) → ran 𝐹 ⊆ (𝐴 ·𝑜 𝐵))
113	30, 112	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (𝐴 ·𝑜 𝐵))
114	111, 113	syl5eqssr 3613	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → dom ◡𝐹 ⊆ (𝐴 ·𝑜 𝐵))
115		relssres 5357	. . . . 5 ⊢ ((Rel ◡𝐹 ∧ dom ◡𝐹 ⊆ (𝐴 ·𝑜 𝐵)) → (◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) = ◡𝐹)
116	110, 114, 115	sylancr 694	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) = ◡𝐹)
117	116	fneq1d 5895	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵) ↔ ◡𝐹 Fn (𝐴 ·𝑜 𝐵)))
118	109, 117	mpbid 221	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡𝐹 Fn (𝐴 ·𝑜 𝐵))
119		dff1o4 6058	. 2 ⊢ (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵) ↔ (𝐹 Fn (𝐵 × 𝐴) ∧ ◡𝐹 Fn (𝐴 ·𝑜 𝐵)))
120	32, 118, 119	sylanbrc 695	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458   ≠ wne 2780  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   class class class wbr 4583  {copab 4642   × cxp 5036  ^◡ccnv 5037  dom cdm 5038  ran crn 5039   ↾ cres 5040  Rel wrel 5043  Ord word 5639  Oncon0 5640  suc csuc 5642   Fn wfn 5799  ⟶wf 5800  –1-1-onto→wf1o 5803  (class class class)co 6549  {coprab 6550   ↦ cmpt2 6551   +_𝑜 coa 7444   ·_𝑜 comu 7445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452

This theorem is referenced by:  omxpen  7947  omf1o  7948  infxpenc  8724