Theorem omxpenlem 8061

Description: Lemma for omxpen 8062. (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 5733	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Ord 𝐵)
2	1	ad2antlr 763	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → Ord 𝐵)
3		simprl 794	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐵)
4		ordsucss 7018	. . . . . . . 8 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → suc 𝑥 ⊆ 𝐵))
5	2, 3, 4	sylc 65	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → suc 𝑥 ⊆ 𝐵)
6		onelon 5748	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
7	6	ad2ant2lr 784	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ On)
8		suceloni 7013	. . . . . . . . 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 792	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝐵 ∈ On)
11		simpll 790	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝐴 ∈ On)
12		omwordi 7651	. . . . . . . 8 ⊢ ((suc 𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑥 ⊆ 𝐵 → (𝐴 ·𝑜 suc 𝑥) ⊆ (𝐴 ·𝑜 𝐵)))
13	9, 10, 11, 12	syl3anc 1326	. . . . . . 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 796	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
16		onelon 5748	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
17	16	ad2ant2rl 785	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ On)
18		omcl 7616	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
19	11, 7, 18	syl2anc 693	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·𝑜 𝑥) ∈ On)
20		oaord 7627	. . . . . . . . 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 1326	. . . . . . . 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 222	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
23		omsuc 7606	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
24	11, 7, 23	syl2anc 693	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·𝑜 suc 𝑥) = ((𝐴 ·𝑜 𝑥) +𝑜 𝐴))
25	22, 24	eleqtrrd 2704	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 suc 𝑥))
26	14, 25	sseldd 3604	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
27	26	ralrimivva 2971	. . . 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 7237	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ↔ 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵))
30	27, 29	sylib 208	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·𝑜 𝐵))
31		ffn 6045	. . 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 790	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝐴 ∈ On)
34		omcl 7616	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
35		onelon 5748	. . . . . . . 8 ⊢ (((𝐴 ·𝑜 𝐵) ∈ On ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝑚 ∈ On)
36	34, 35	sylan 488	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝑚 ∈ On)
37		noel 3919	. . . . . . . . . . . 12 ⊢ ¬ 𝑚 ∈ ∅
38		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
39		om0r 7619	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
40	38, 39	sylan9eqr 2678	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
41	40	eleq2d 2687	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) ↔ 𝑚 ∈ ∅))
42	37, 41	mtbiri 317	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ¬ 𝑚 ∈ (𝐴 ·𝑜 𝐵))
43	42	ex 450	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (𝐴 = ∅ → ¬ 𝑚 ∈ (𝐴 ·𝑜 𝐵)))
44	43	necon2ad 2809	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝑚 ∈ (𝐴 ·𝑜 𝐵) → 𝐴 ≠ ∅))
45	44	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) → 𝐴 ≠ ∅))
46	45	imp 445	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → 𝐴 ≠ ∅)
47		omeu 7665	. . . . . . 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 1326	. . . . . 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 3203	. . . . . . . . 9 ⊢ 𝑚 ∈ V
50		vex 3203	. . . . . . . . 9 ⊢ 𝑛 ∈ V
51	49, 50	brcnv 5305	. . . . . . . 8 ⊢ (𝑚◡𝐹𝑛 ↔ 𝑛𝐹𝑚)
52		eleq1 2689	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) → (𝑚 ∈ (𝐴 ·𝑜 𝐵) ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)))
53	52	biimpac 503	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)) → ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵))
54	6	ex 450	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∈ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
55	54	ad2antlr 763	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
56		simplll 798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
57		simpr 477	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
58	56, 57, 18	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝑥) ∈ On)
59		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴)
60	56, 59, 16	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
61		oaword1 7632	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ·𝑜 𝑥) ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·𝑜 𝑥) ⊆ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
62	58, 60, 61	syl2anc 693	. . . . . . . . . . . . . . . . . . . . . 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 800	. . . . . . . . . . . . . . . . . . . . . 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 762	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·𝑜 𝐵) ∈ On)
65		ontr2 5772	. . . . . . . . . . . . . . . . . . . . . . 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 693	. . . . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
69		ne0i 3921	. . . . . . . . . . . . . . . . . . . . . . . . . 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 5780	. . . . . . . . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ (𝐴 ·𝑜 𝐵))
74		om00el 7656	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·𝑜 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
75	74	ad2antrr 762	. . . . . . . . . . . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
77	76	simpld 475	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ 𝐴)
78		omord2 7647	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ (𝐴 ·𝑜 𝑥) ∈ (𝐴 ·𝑜 𝐵)))
79	57, 68, 56, 77, 78	syl31anc 1329	. . . . . . . . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ 𝐵)
81	80	ex 450	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐵))
82	55, 81	impbid 202	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ On))
83	82	expr 643	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ∈ (𝐴 ·𝑜 𝐵)) → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ On)))
84	83	pm5.32rd 672	. . . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . 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 672	. . . . . . . . . . . . 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 2629	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) ↔ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦) = 𝑚)
89	88	anbi2i 730	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 740	. . . . . . . . . . 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 838	. . . . . . . . . . 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 276	. . . . . . . . . 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 1852	. . . . . . . . 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 6655	. . . . . . . . . . . 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 6701	. . . . . . . . . . . 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 2648	. . . . . . . . . . 11 ⊢ 𝐹 = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}
98	97	breqi 4659	. . . . . . . . . 10 ⊢ (𝑛𝐹𝑚 ↔ 𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦)))}𝑚)
99		df-br 4654	. . . . . . . . . 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 4982	. . . . . . . . . 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 286	. . . . . . . . 9 ⊢ (𝑛𝐹𝑚 ↔ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))))
102		r2ex 3061	. . . . . . . . 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 303	. . . . . . . 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 272	. . . . . . 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 2490	. . . . . 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 247	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·𝑜 𝐵)) → ∃!𝑛 𝑚◡𝐹𝑛)
107	106	ralrimiva 2966	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑚 ∈ (𝐴 ·𝑜 𝐵)∃!𝑛 𝑚◡𝐹𝑛)
108		fnres 6007	. . . 4 ⊢ ((◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵) ↔ ∀𝑚 ∈ (𝐴 ·𝑜 𝐵)∃!𝑛 𝑚◡𝐹𝑛)
109	107, 108	sylibr 224	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵))
110		relcnv 5503	. . . . 5 ⊢ Rel ◡𝐹
111		df-rn 5125	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
112		frn 6053	. . . . . . 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 3650	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → dom ◡𝐹 ⊆ (𝐴 ·𝑜 𝐵))
115		relssres 5437	. . . . 5 ⊢ ((Rel ◡𝐹 ∧ dom ◡𝐹 ⊆ (𝐴 ·𝑜 𝐵)) → (◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) = ◡𝐹)
116	110, 114, 115	sylancr 695	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) = ◡𝐹)
117	116	fneq1d 5981	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((◡𝐹 ↾ (𝐴 ·𝑜 𝐵)) Fn (𝐴 ·𝑜 𝐵) ↔ ◡𝐹 Fn (𝐴 ·𝑜 𝐵)))
118	109, 117	mpbid 222	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡𝐹 Fn (𝐴 ·𝑜 𝐵))
119		dff1o4 6145	. 2 ⊢ (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵) ↔ (𝐹 Fn (𝐵 × 𝐴) ∧ ◡𝐹 Fn (𝐴 ·𝑜 𝐵)))
120	32, 118, 119	sylanbrc 698	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183   class class class wbr 4653  {copab 4712   × cxp 5112  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116  Rel wrel 5119  Ord word 5722  Oncon0 5723  suc csuc 5725   Fn wfn 5883  ⟶wf 5884  –1-1-onto→wf1o 5887  (class class class)co 6650  {coprab 6651   ↦ cmpt2 6652   +_𝑜 coa 7557   ·_𝑜 comu 7558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565

This theorem is referenced by:  omxpen  8062  omf1o  8063  infxpenc  8841