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Theorem oacomf1o 7803
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 8716). (Contributed by Mario Carneiro, 30-May-2015.)
Hypothesis
Ref Expression
oacomf1o.1 𝐹 = ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ (𝑥𝐵 ↦ (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
StepHypRef Expression
1 eqid 2771 . . . . . . 7 (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) = (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))
21oacomf1olem 7802 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∧ (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅))
32simpld 482 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)))
4 eqid 2771 . . . . . . . . 9 (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
54oacomf1olem 7802 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅))
65ancoms 446 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅))
76simpld 482 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))
8 f1ocnv 6291 . . . . . 6 ((𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) → (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))–1-1-onto𝐵)
97, 8syl 17 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))–1-1-onto𝐵)
10 incom 3956 . . . . . 6 (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))) = (ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴)
116simprd 483 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅)
1210, 11syl5eq 2817 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))) = ∅)
132simprd 483 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅)
14 f1oun 6298 . . . . 5 ((((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∧ (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))–1-1-onto𝐵) ∧ ((𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))) = ∅ ∧ (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅)) → ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))):(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
153, 9, 12, 13, 14syl22anc 1477 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))):(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
16 oacomf1o.1 . . . . 5 𝐹 = ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))
17 f1oeq1 6269 . . . . 5 (𝐹 = ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))) → (𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) ↔ ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))):(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
1816, 17ax-mp 5 . . . 4 (𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) ↔ ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))):(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
1915, 18sylibr 224 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
20 oarec 7800 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))))
21 f1oeq2 6270 . . . 4 ((𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))) → (𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) ↔ 𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
2220, 21syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) ↔ 𝐹:(𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
2319, 22mpbird 247 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
24 oarec 7800 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +𝑜 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))))
2524ancoms 446 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))))
26 uncom 3908 . . . 4 (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))) = (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)
2725, 26syl6eq 2821 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝐴) = (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
28 f1oeq3 6271 . . 3 ((𝐵 +𝑜 𝐴) = (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵) → (𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴) ↔ 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
2927, 28syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴) ↔ 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)))
3023, 29mpbird 247 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 +𝑜 𝐵)–1-1-onto→(𝐵 +𝑜 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  cun 3721  cin 3722  c0 4063  cmpt 4864  ccnv 5249  ran crn 5251  Oncon0 5865  1-1-ontowf1o 6029  (class class class)co 6796   +𝑜 coa 7714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-oadd 7721
This theorem is referenced by:  cnfcomlem  8764
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