Theorem oacomf1o 8396

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 9409). (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 2738	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))
2	1	oacomf1olem 8395	. . . . . 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 495	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)))
4		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
5	4	oacomf1olem 8395	. . . . . . . 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 459	. . . . . . 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 495	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
8		f1ocnv 6728	. . . . . 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 4135	. . . . . 6 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
11	6	simprd 496	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	eqtrid 2790	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13	2	simprd 496	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)
14		f1oun 6735	. . . . 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 836	. . . 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 6704	. . . . 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 8393	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
21	20	f1oeq2d 6712	. . 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 8393	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
24	23	ancoms 459	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
25		uncom 4087	. . . 4 ⊢ (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)
26	24, 25	eqtrdi 2794	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
27	26	f1oeq3d 6713	. 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 396   = wceq 1539   ∈ wcel 2106   ∪ cun 3885   ∩ cin 3886  ∅c0 4256   ↦ cmpt 5157  ^◡ccnv 5588  ran crn 5590  Oncon0 6266  –1-1-onto→wf1o 6432  (class class class)co 7275   +_o coa 8294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-oadd 8301

This theorem is referenced by:  cnfcomlem  9457