Theorem oacomf1o 8611

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 9689). (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)⟶1-1onto(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 2763	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) = (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))
2	1	oacomf1olem 8610	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)):𝐴⟶1-1ontoran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∧ (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅))
3	2	simpld 500	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)):𝐴⟶1-1ontoran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)))
4		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) = (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))
5	4	oacomf1olem 8610	. . . . . . . 8 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵⟶1-1ontoran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
6	5	ancoms 464	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵⟶1-1ontoran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∧ (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅))
7	6	simpld 500	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵⟶1-1ontoran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
8		f1ocnv 6867	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):𝐵⟶1-1ontoran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) → ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))⟶1-1onto𝐵)
9	7, 8	syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))⟶1-1onto𝐵)
10		incom 4109	. . . . . 6 ⊢ (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴)
11	6	simprd 501	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)) ∩ 𝐴) = ∅)
12	10, 11	eqtrid 2810	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) = ∅)
13	2	simprd 501	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∩ 𝐵) = ∅)
14		f1oun 6874	. . . . 5 ⊢ ((((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)):𝐴⟶1-1ontoran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∧ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)):ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))⟶1-1onto𝐵) ∧ ((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)))⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
15	3, 9, 12, 13, 14	syl22anc 852	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))):(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
16		oacomf1o.1	. . . . 5 ⊢ 𝐹 = ((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))
17		f1oeq1 6842	. . . . 5 ⊢ (𝐹 = ((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))) → (𝐹:(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵) ↔ ((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))):(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)))
18	16, 17	ax-mp 5	. . . 4 ⊢ (𝐹:(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵) ↔ ((𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ ◡(𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))):(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))⟶1-1onto(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)))⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
20		oarec 8608	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +o 𝐵) = (𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥))))
21	20	f1oeq2d 6850	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:(𝐴 +o 𝐵)⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵) ↔ 𝐹:(𝐴 ∪ ran (𝑥 ∈ 𝐵 ↦ (𝐴 +o 𝑥)))⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)))
22	19, 21	mpbird 260	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
23		oarec 8608	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
24	23	ancoms 464	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))))
25		uncom 4160	. . . 4 ⊢ (𝐵 ∪ ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥))) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)
26	24, 25	eqtrdi 2814	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +o 𝐴) = (ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵))
27	26	f1oeq3d 6851	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:(𝐴 +o 𝐵)⟶1-1onto(𝐵 +o 𝐴) ↔ 𝐹:(𝐴 +o 𝐵)⟶1-1onto(ran (𝑥 ∈ 𝐴 ↦ (𝐵 +o 𝑥)) ∪ 𝐵)))
28	22, 27	mpbird 260	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +o 𝐵)⟶1-1onto(𝐵 +o 𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ∩ cin 4092   ∪ cun 4152  ∅c0 4281   ↦ cmpt 5192  ^◡ccnv 5657  ran crn 5659  Oncon0 6388  ⟶1-1ontowf1o 6565  (class class class)co 7451   +_o coa 8511

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-oadd 8518

This theorem is used by:  cnfcomlem  9737