Theorem oacomf1o 7645

Description: Define a bijection from 𝐴 +_𝑜 𝐵 to 𝐵 +_𝑜 𝐴. Thus, the two are equinumerous even if they are not equal (which sometimes occurs, e.g. oancom 8548). (Contributed by Mario Carneiro, 30-May-2015.)

Hypothesis

Ref	Expression
oacomf1o.1	⊢ 𝐹 = ((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))

Assertion

Ref	Expression
oacomf1o	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem oacomf1o

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))
2	1	oacomf1olem 7644	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∧ (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅))
3	2	simpld 475	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)))
4		eqid 2622	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))
5	4	oacomf1olem 7644	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∧ (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅))
6	5	ancoms 469	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∧ (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅))
7	6	simpld 475	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
8		f1ocnv 6149	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) → ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))–1-1-onto→𝐵)
9	7, 8	syl 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))–1-1-onto→𝐵)
10		incom 3805	. . . . . 6 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴)
11	6	simprd 479	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	syl5eq 2668	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) = ∅)
13	2	simprd 479	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅)
14		f1oun 6156	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∧ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))–1-1-onto→𝐵) ∧ ((𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) = ∅ ∧ (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅)) → ((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))):(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
15	3, 9, 12, 13, 14	syl22anc 1327	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))):(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
16		oacomf1o.1	. . . . 5 ⊢ 𝐹 = ((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
17		f1oeq1 6127	. . . . 5 ⊢ (𝐹 = ((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) → (𝐹:(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) ↔ ((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))):(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
18	16, 17	ax-mp 5	. . . 4 ⊢ (𝐹:(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) ↔ ((𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))):(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
19	15, 18	sylibr 224	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
20		oarec 7642	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))
21		f1oeq2 6128	. . . 4 ⊢ ((𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) → (𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) ↔ 𝐹:(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
22	20, 21	syl 17	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) ↔ 𝐹:(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
23	19, 22	mpbird 247	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
24		oarec 7642	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +𝑜 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))))
25	24	ancoms 469	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))))
26		uncom 3757	. . . 4 ⊢ (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)
27	25, 26	syl6eq 2672	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝐴) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
28		f1oeq3 6129	. . 3 ⊢ ((𝐵 +𝑜 𝐴) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) → (𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴) ↔ 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
29	27, 28	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴) ↔ 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
30	23, 29	mpbird 247	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   ∩ cin 3573  ∅c0 3915   ↦ cmpt 4729  ^◡ccnv 5113  ran crn 5115  Oncon0 5723  –1-1-onto→wf1o 5887  (class class class)co 6650   +_𝑜 coa 7557

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-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  cnfcomlem  8596