Theorem omxpenlem 8351

Description: Lemma for omxpen 8352. (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 5988	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Ord 𝐵)
2	1	ad2antlr 717	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → Ord 𝐵)
3		simprl 761	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐵)
4		ordsucss 7298	. . . . . . . 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 6003	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
7	6	ad2ant2lr 738	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ On)
8		suceloni 7293	. . . . . . . . 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 759	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝐵 ∈ On)
11		simpll 757	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝐴 ∈ On)
12		omwordi 7937	. . . . . . . 8 ⊢ ((suc 𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑥 ⊆ 𝐵 → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵)))
13	9, 10, 11, 12	syl3anc 1439	. . . . . . 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 763	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
16		onelon 6003	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
17	16	ad2ant2rl 739	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ On)
18		omcl 7902	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
19	11, 7, 18	syl2anc 579	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·o 𝑥) ∈ On)
20		oaord 7913	. . . . . . . . 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 1439	. . . . . . . 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 224	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴))
23		omsuc 7892	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
24	11, 7, 23	syl2anc 579	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
25	22, 24	eleqtrrd 2862	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o suc 𝑥))
26	14, 25	sseldd 3822	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
27	26	ralrimivva 3153	. . . 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 7519	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ↔ 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
30	27, 29	sylib 210	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
31	30	ffnd 6294	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐵 × 𝐴))
32		simpll 757	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ∈ On)
33		omcl 7902	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
34		onelon 6003	. . . . . . . 8 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
35	33, 34	sylan 575	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
36		noel 4146	. . . . . . . . . . . 12 ⊢ ¬ 𝑚 ∈ ∅
37		oveq1 6931	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
38		om0r 7905	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
39	37, 38	sylan9eqr 2836	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
40	39	eleq2d 2845	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝑚 ∈ (𝐴 ·o 𝐵) ↔ 𝑚 ∈ ∅))
41	36, 40	mtbiri 319	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ¬ 𝑚 ∈ (𝐴 ·o 𝐵))
42	41	ex 403	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (𝐴 = ∅ → ¬ 𝑚 ∈ (𝐴 ·o 𝐵)))
43	42	necon2ad 2984	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝑚 ∈ (𝐴 ·o 𝐵) → 𝐴 ≠ ∅))
44	43	adantl 475	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑚 ∈ (𝐴 ·o 𝐵) → 𝐴 ≠ ∅))
45	44	imp 397	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ≠ ∅)
46		omeu 7951	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝑚 ∈ On ∧ 𝐴 ≠ ∅) → ∃!𝑛∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
47	32, 35, 45, 46	syl3anc 1439	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ∃!𝑛∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
48		vex 3401	. . . . . . . . 9 ⊢ 𝑚 ∈ V
49		vex 3401	. . . . . . . . 9 ⊢ 𝑛 ∈ V
50	48, 49	brcnv 5552	. . . . . . . 8 ⊢ (𝑚◡𝐹𝑛 ↔ 𝑛𝐹𝑚)
51		eleq1 2847	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) → (𝑚 ∈ (𝐴 ·o 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)))
52	51	biimpac 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (𝐴 ·o 𝐵) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
53	6	ex 403	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∈ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
54	53	ad2antlr 717	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
55		simplll 765	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
56		simpr 479	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
57	55, 56, 18	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
58		simplrr 768	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴)
59	55, 58, 16	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
60		oaword1 7918	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ·o 𝑥) ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑥) ⊆ ((𝐴 ·o 𝑥) +o 𝑦))
61	57, 59, 60	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . 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))
62		simplrl 767	. . . . . . . . . . . . . . . . . . . . . 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))
63	33	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
64		ontr2 6025	. . . . . . . . . . . . . . . . . . . . . . 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)))
65	57, 63, 64	syl2anc 579	. . . . . . . . . . . . . . . . . . . . . 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)))
66	61, 62, 65	mp2and 689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵))
67		simpllr 766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
68	62	ne0d 4150	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ≠ ∅)
69		on0eln0 6033	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ·o 𝐵) ∈ On → (∅ ∈ (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) ↔ (𝐴 ·o 𝐵) ≠ ∅))
71	68, 70	mpbird 249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ (𝐴 ·o 𝐵))
72		om00el 7942	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
73	72	ad2antrr 716	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
74	71, 73	mpbid 224	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
75	74	simpld 490	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ 𝐴)
76		omord2 7933	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵)))
77	56, 67, 55, 75, 76	syl31anc 1441	. . . . . . . . . . . . . . . . . . . . 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)))
78	66, 77	mpbird 249	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ 𝐵)
79	78	ex 403	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐵))
80	54, 79	impbid 204	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ On))
81	80	expr 450	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ On)))
82	81	pm5.32rd 573	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ 𝐴)))
83	52, 82	sylan2 586	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑚 ∈ (𝐴 ·o 𝐵) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ 𝐴)))
84	83	expr 450	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ On ∧ 𝑦 ∈ 𝐴))))
85	84	pm5.32rd 573	. . . . . . . . . . . . 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))))
86		eqcom 2785	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)
87	86	anbi2i 616	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
88	85, 87	syl6bb 279	. . . . . . . . . . . 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)))
89	88	anbi2d 622	. . . . . . . . . . 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))))
90		an12 635	. . . . . . . . . . 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)))
91	89, 90	syl6bb 279	. . . . . . . . . 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))))
92	91	2exbidv 1967	. . . . . . . . 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))))
93		df-mpt2 6929	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦)) = {⟨⟨𝑥, 𝑦⟩, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))}
94		dfoprab2 6980	. . . . . . . . . . . 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)))}
95	28, 93, 94	3eqtri 2806	. . . . . . . . . . 11 ⊢ 𝐹 = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}
96	95	breqi 4894	. . . . . . . . . 10 ⊢ (𝑛𝐹𝑚 ↔ 𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}𝑚)
97		df-br 4889	. . . . . . . . . 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)))})
98		opabid 5221	. . . . . . . . . 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))))
99	96, 97, 98	3bitri 289	. . . . . . . . 9 ⊢ (𝑛𝐹𝑚 ↔ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))))
100		r2ex 3246	. . . . . . . . 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)))
101	92, 99, 100	3bitr4g 306	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑛𝐹𝑚 ↔ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
102	50, 101	syl5bb 275	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → (𝑚◡𝐹𝑛 ↔ ∃𝑥 ∈ On ∃𝑦 ∈ 𝐴 (𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)))
103	102	eubidv 2606	. . . . . 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)))
104	47, 103	mpbird 249	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ∃!𝑛 𝑚◡𝐹𝑛)
105	104	ralrimiva 3148	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚◡𝐹𝑛)
106		fnres 6255	. . . 4 ⊢ ((◡𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚◡𝐹𝑛)
107	105, 106	sylibr 226	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵))
108		relcnv 5759	. . . . 5 ⊢ Rel ◡𝐹
109		df-rn 5368	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
110	30	frnd 6300	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (𝐴 ·o 𝐵))
111	109, 110	syl5eqssr 3869	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → dom ◡𝐹 ⊆ (𝐴 ·o 𝐵))
112		relssres 5689	. . . . 5 ⊢ ((Rel ◡𝐹 ∧ dom ◡𝐹 ⊆ (𝐴 ·o 𝐵)) → (◡𝐹 ↾ (𝐴 ·o 𝐵)) = ◡𝐹)
113	108, 111, 112	sylancr 581	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝐹 ↾ (𝐴 ·o 𝐵)) = ◡𝐹)
114	113	fneq1d 6228	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((◡𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ ◡𝐹 Fn (𝐴 ·o 𝐵)))
115	107, 114	mpbid 224	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡𝐹 Fn (𝐴 ·o 𝐵))
116		dff1o4 6401	. 2 ⊢ (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵) ↔ (𝐹 Fn (𝐵 × 𝐴) ∧ ◡𝐹 Fn (𝐴 ·o 𝐵)))
117	31, 115, 116	sylanbrc 578	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∃!weu 2586   ≠ wne 2969  ∀wral 3090  ∃wrex 3091   ⊆ wss 3792  ∅c0 4141  ⟨cop 4404   class class class wbr 4888  {copab 4950   × cxp 5355  ^◡ccnv 5356  dom cdm 5357  ran crn 5358   ↾ cres 5359  Rel wrel 5362  Ord word 5977  Oncon0 5978  suc csuc 5980   Fn wfn 6132  ⟶wf 6133  –1-1-onto→wf1o 6136  (class class class)co 6924  {coprab 6925   ↦ cmpt2 6926   +_o coa 7842   ·_o comu 7843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4674  df-int 4713  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-tr 4990  df-id 5263  df-eprel 5268  df-po 5276  df-so 5277  df-fr 5316  df-we 5318  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-pred 5935  df-ord 5981  df-on 5982  df-lim 5983  df-suc 5984  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-om 7346  df-1st 7447  df-2nd 7448  df-wrecs 7691  df-recs 7753  df-rdg 7791  df-1o 7845  df-oadd 7849  df-omul 7850

This theorem is referenced by:  omxpen  8352  omf1o  8353  infxpenc  9176