Theorem oacomf1o 8549

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 9619). (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 2769	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))
2	1	oacomf1olem 8548	. . . . . 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 499	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)))
4		eqid 2769	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
5	4	oacomf1olem 8548	. . . . . . . 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 463	. . . . . . 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 499	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
8		f1ocnv 6834	. . . . . 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 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))–1-1-onto→𝐵)
10		incom 4170	. . . . . 6 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
11	6	simprd 500	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	eqtrid 2816	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13	2	simprd 500	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)
14		f1oun 6841	. . . . 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 851	. . . 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 6809	. . . . 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 237	. . 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 8546	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
21	20	f1oeq2d 6817	. . 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 260	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
23		oarec 8546	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
24	23	ancoms 463	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
25		uncom 4120	. . . 4 ⊢ (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)
26	24, 25	eqtrdi 2820	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
27	26	f1oeq3d 6818	. 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 260	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149   ∪ cun 3911   ∩ cin 3912  ∅c0 4294   ↦ cmpt 5196  ^◡ccnv 5661  ran crn 5663  Oncon0 6361  –1-1-onto→wf1o 6536  (class class class)co 7411   +_o coa 8449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-oadd 8456

This theorem is referenced by:  cnfcomlem  9667