Theorem oacomf1o 8193

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 9116). (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 2823	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))
2	1	oacomf1olem 8192	. . . . . 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 497	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)):𝐴–1-1-onto→ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)))
4		eqid 2823	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
5	4	oacomf1olem 8192	. . . . . . . 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 461	. . . . . . 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 497	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵–1-1-onto→ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
8		f1ocnv 6629	. . . . . 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 4180	. . . . . 6 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
11	6	simprd 498	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	syl5eq 2870	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13	2	simprd 498	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)
14		f1oun 6636	. . . . 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 6606	. . . . 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 236	. . 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 8190	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
21	20	f1oeq2d 6613	. . 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 259	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
23		oarec 8190	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
24	23	ancoms 461	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
25		uncom 4131	. . . 4 ⊢ (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)
26	24, 25	syl6eq 2874	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
27	26	f1oeq3d 6614	. 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 259	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)–1-1-onto→(𝐵 +o 𝐴))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ∪ cun 3936   ∩ cin 3937  ∅c0 4293   ↦ cmpt 5148  ^◡ccnv 5556  ran crn 5558  Oncon0 6193  –1-1-onto→wf1o 6356  (class class class)co 7158   +_o coa 8101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-oadd 8108

This theorem is referenced by:  cnfcomlem  9164