Theorem omxpenlem 9097

Description: Lemma for omxpen 9098. (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 6372	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Ord 𝐵)
2	1	ad2antlr 740	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → Ord 𝐵)
3		simprl 783	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ 𝐵)
4		ordsucss 7829	. . . . . . . 8 ⊢ (Ord 𝐵 → (𝑥 ∈ 𝐵 → suc 𝑥 ⊆ 𝐵))
5	2, 3, 4	sylc 66	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → suc 𝑥 ⊆ 𝐵)
6		onelon 6387	. . . . . . . . . 10 ⊢ ((𝐵 ∈ On ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ On)
7	6	ad2ant2lr 761	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ On)
8		onsuc 7824	. . . . . . . . 9 ⊢ (𝑥 ∈ On → suc 𝑥 ∈ On)
9	7, 8	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → suc 𝑥 ∈ On)
10		simplr 781	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝐵 ∈ On)
11		simpll 779	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝐴 ∈ On)
12		omwordi 8579	. . . . . . . 8 ⊢ ((suc 𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (suc 𝑥 ⊆ 𝐵 → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵)))
13	9, 10, 11, 12	syl3anc 1398	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (suc 𝑥 ⊆ 𝐵 → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵)))
14	5, 13	mpd 16	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·o suc 𝑥) ⊆ (𝐴 ·o 𝐵))
15		simprr 785	. . . . . . . 8 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴)
16		onelon 6387	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
17	16	ad2ant2rl 762	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ On)
18		omcl 8544	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
19	11, 7, 18	syl2anc 596	. . . . . . . . 9 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·o 𝑥) ∈ On)
20		oaord 8555	. . . . . . . . 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 1398	. . . . . . . 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 235	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ ((𝐴 ·o 𝑥) +o 𝐴))
23		omsuc 8534	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝑥 ∈ On) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
24	11, 7, 23	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → (𝐴 ·o suc 𝑥) = ((𝐴 ·o 𝑥) +o 𝐴))
25	22, 24	eleqtrrd 2864	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o suc 𝑥))
26	14, 25	sseldd 3932	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
27	26	ralrimivva 3206	. . . 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	fmpo 8079	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐴 ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ↔ 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
30	27, 29	sylib 221	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)⟶(𝐴 ·o 𝐵))
31	30	ffnd 6710	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹 Fn (𝐵 × 𝐴))
32		simpll 779	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ∈ On)
33		omcl 8544	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
34		onelon 6387	. . . . . . . 8 ⊢ (((𝐴 ·o 𝐵) ∈ On ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
35	33, 34	sylan 592	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝑚 ∈ On)
36		noel 4284	. . . . . . . . . . . 12 ⊢ ¬ 𝑚 ∈ ∅
37		oveq1 7427	. . . . . . . . . . . . . 14 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
38		om0r 8547	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
39	37, 38	sylan9eqr 2818	. . . . . . . . . . . . 13 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
40	39	eleq2d 2847	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → (𝑚 ∈ (𝐴 ·o 𝐵) ↔ 𝑚 ∈ ∅))
41	36, 40	mtbiri 330	. . . . . . . . . . 11 ⊢ ((𝐵 ∈ On ∧ 𝐴 = ∅) → ¬ 𝑚 ∈ (𝐴 ·o 𝐵))
42	41	ex 418	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → (𝐴 = ∅ → ¬ 𝑚 ∈ (𝐴 ·o 𝐵)))
43	42	necon2ad 2971	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (𝑚 ∈ (𝐴 ·o 𝐵) → 𝐴 ≠ ∅))
44	43	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑚 ∈ (𝐴 ·o 𝐵) → 𝐴 ≠ ∅))
45	44	imp 412	. . . . . . 7 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → 𝐴 ≠ ∅)
46		omeu 8593	. . . . . . 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 1398	. . . . . 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 3455	. . . . . . . . 9 ⊢ 𝑚 ∈ V
49		vex 3455	. . . . . . . . 9 ⊢ 𝑛 ∈ V
50	48, 49	brcnv 5860	. . . . . . . 8 ⊢ (𝑚◡𝐹𝑛 ↔ 𝑛𝐹𝑚)
51		eleq1 2849	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) → (𝑚 ∈ (𝐴 ·o 𝐵) ↔ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)))
52	51	biimpac 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝑚 ∈ (𝐴 ·o 𝐵) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) → ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵))
53	6	ex 418	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐵 ∈ On → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
54	53	ad2antlr 740	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ On))
55		simplll 787	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝐴 ∈ On)
56		simpr 490	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ On)
57	55, 56, 18	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝑥) ∈ On)
58		simplrr 790	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ 𝐴)
59	55, 58, 16	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑦 ∈ On)
60		oaword1 8560	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐴 ·o 𝑥) ∈ On ∧ 𝑦 ∈ On) → (𝐴 ·o 𝑥) ⊆ ((𝐴 ·o 𝑥) +o 𝑦))
61	57, 59, 60	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . 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 789	. . . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ∈ On)
64		ontr2 6411	. . . . . . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . . . . . 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 712	. . . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝐵 ∈ On)
68	62	ne0d 4288	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (𝐴 ·o 𝐵) ≠ ∅)
69		on0eln0 6420	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝐴 ·o 𝐵) ∈ On → (∅ ∈ (𝐴 ·o 𝐵) ↔ (𝐴 ·o 𝐵) ≠ ∅))
70	63, 69	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ (𝐴 ·o 𝐵))
72		om00el 8584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ (𝐴 ·o 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)))
73	72	ad2antrr 739	. . . . . . . . . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
75	74	simpld 500	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → ∅ ∈ 𝐴)
76		omord2 8575	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑥 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) ∧ ∅ ∈ 𝐴) → (𝑥 ∈ 𝐵 ↔ (𝐴 ·o 𝑥) ∈ (𝐴 ·o 𝐵)))
77	56, 67, 55, 75, 76	syl31anc 1400	. . . . . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) ∧ 𝑥 ∈ On) → 𝑥 ∈ 𝐵)
79	78	ex 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ On → 𝑥 ∈ 𝐵))
80	54, 79	impbid 215	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ (((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵) ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ On))
81	80	expr 462	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ((𝐴 ·o 𝑥) +o 𝑦) ∈ (𝐴 ·o 𝐵)) → (𝑦 ∈ 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ On)))
82	81	pm5.32rd 589	. . . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . . 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 462	. . . . . . . . . . . . . 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 589	. . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ((𝐴 ·o 𝑥) +o 𝑦) ↔ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚)
87	86	anbi2i 635	. . . . . . . . . . . . 13 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ ((𝐴 ·o 𝑥) +o 𝑦) = 𝑚))
88	85, 87	bitrdi 290	. . . . . . . . . . . 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 642	. . . . . . . . . . 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 658	. . . . . . . . . . 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	bitrdi 290	. . . . . . . . . 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 1957	. . . . . . . . 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-mpo 7425	. . . . . . . . . . . 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 7478	. . . . . . . . . . . 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 2788	. . . . . . . . . . 11 ⊢ 𝐹 = {⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}
96	95	breqi 5109	. . . . . . . . . 10 ⊢ (𝑛𝐹𝑚 ↔ 𝑛{⟨𝑛, 𝑚⟩ ∣ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦)))}𝑚)
97		df-br 5104	. . . . . . . . . 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		opabidw 5498	. . . . . . . . . 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 300	. . . . . . . . 9 ⊢ (𝑛𝐹𝑚 ↔ ∃𝑥∃𝑦(𝑛 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐴) ∧ 𝑚 = ((𝐴 ·o 𝑥) +o 𝑦))))
100		r2ex 3200	. . . . . . . . 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 317	. . . . . . . 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	bitrid 286	. . . . . . 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 2612	. . . . . 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 260	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑚 ∈ (𝐴 ·o 𝐵)) → ∃!𝑛 𝑚◡𝐹𝑛)
105	104	ralrimiva 3155	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚◡𝐹𝑛)
106		fnres 6666	. . . 4 ⊢ ((◡𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ ∀𝑚 ∈ (𝐴 ·o 𝐵)∃!𝑛 𝑚◡𝐹𝑛)
107	105, 106	sylibr 237	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵))
108		relcnv 6100	. . . . 5 ⊢ Rel ◡𝐹
109		df-rn 5662	. . . . . 6 ⊢ ran 𝐹 = dom ◡𝐹
110	30	frnd 6718	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ran 𝐹 ⊆ (𝐴 ·o 𝐵))
111	109, 110	eqsstrrid 3970	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → dom ◡𝐹 ⊆ (𝐴 ·o 𝐵))
112		relssres 6011	. . . . 5 ⊢ ((Rel ◡𝐹 ∧ dom ◡𝐹 ⊆ (𝐴 ·o 𝐵)) → (◡𝐹 ↾ (𝐴 ·o 𝐵)) = ◡𝐹)
113	108, 111, 112	sylancr 599	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (◡𝐹 ↾ (𝐴 ·o 𝐵)) = ◡𝐹)
114	113	fneq1d 6632	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((◡𝐹 ↾ (𝐴 ·o 𝐵)) Fn (𝐴 ·o 𝐵) ↔ ◡𝐹 Fn (𝐴 ·o 𝐵)))
115	107, 114	mpbid 235	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡𝐹 Fn (𝐴 ·o 𝐵))
116		dff1o4 6833	. 2 ⊢ (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵) ↔ (𝐹 Fn (𝐵 × 𝐴) ∧ ◡𝐹 Fn (𝐴 ·o 𝐵)))
117	31, 115, 116	sylanbrc 595	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2594   ≠ wne 2956  ∀wral 3077  ∃wrex 3087   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590   class class class wbr 5103  {copab 5167   × cxp 5649  ^◡ccnv 5650  dom cdm 5651  ran crn 5652   ↾ cres 5653  Rel wrel 5656  Ord word 6361  Oncon0 6362  suc csuc 6364   Fn wfn 6533  ⟶wf 6534  –1-1-onto→wf1o 6537  (class class class)co 7420  {coprab 7421   ∈ cmpo 7422   +_o coa 8473   ·_o comu 8474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-oadd 8480  df-omul 8481

This theorem is used by:  omxpen  9098  omf1o  9099  infxpenc  10097