Theorem oacomf1o 8567

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 9648). (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 2730	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))
2	1	oacomf1olem 8566	. . . . . 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 493	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)))
4		eqid 2730	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
5	4	oacomf1olem 8566	. . . . . . . 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 457	. . . . . . 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 493	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
8		f1ocnv 6844	. . . . . 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 4200	. . . . . 6 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
11	6	simprd 494	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	eqtrid 2782	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13	2	simprd 494	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)
14		f1oun 6851	. . . . 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 835	. . . 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 6820	. . . . 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 233	. . 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 8564	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
21	20	f1oeq2d 6828	. . 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)))
22	19, 21	mpbird 256	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
23		oarec 8564	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
24	23	ancoms 457	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
25		uncom 4152	. . . 4 ⊢ (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)
26	24, 25	eqtrdi 2786	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
27	26	f1oeq3d 6829	. 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)))
28	22, 27	mpbird 256	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104   ∪ cun 3945   ∩ cin 3946  ∅c0 4321   ↦ cmpt 5230  ^◡ccnv 5674  ran crn 5676  Oncon0 6363  –1-1-onto→wf1o 6541  (class class class)co 7411   +_o coa 8465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-oadd 8472

This theorem is referenced by:  cnfcomlem  9696