Theorem oacomf1o 7532

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 8431). (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 2610	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))
2	1	oacomf1olem 7531	. . . . . 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 474	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)))
4		eqid 2610	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))
5	4	oacomf1olem 7531	. . . . . . . 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 468	. . . . . . 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 474	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)))
8		f1ocnv 6062	. . . . . 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 3767	. . . . . 6 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴)
11	6	simprd 478	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	syl5eq 2656	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))) = ∅)
13	2	simprd 478	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅)
14		f1oun 6069	. . . . 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 1319	. . . 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 6040	. . . . 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 223	. . 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 7529	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +𝑜 𝑥))))
21		f1oeq2 6041	. . . 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 246	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
24		oarec 7529	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +𝑜 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))))
25	24	ancoms 468	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))))
26		uncom 3719	. . . 4 ⊢ (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥))) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)
27	25, 26	syl6eq 2660	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝐴) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
28		f1oeq3 6042	. . 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 246	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ∩ cin 3539  ∅c0 3874   ↦ cmpt 4643  ^◡ccnv 5037  ran crn 5039  Oncon0 5640  –1-1-onto→wf1o 5803  (class class class)co 6549   +_𝑜 coa 7444

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-wrecs 7294  df-recs 7355  df-rdg 7393  df-oadd 7451

This theorem is referenced by:  cnfcomlem  8479