Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oacomf1o Structured version   Visualization version   GIF version

Theorem oacomf1o 7690
 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 8586). (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 2651 . . . . . . 7 (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) = (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))
21oacomf1olem 7689 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∧ (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅))
32simpld 474 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)):𝐴1-1-onto→ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)))
4 eqid 2651 . . . . . . . . 9 (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) = (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))
54oacomf1olem 7689 . . . . . . . 8 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅))
65ancoms 468 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∧ (ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅))
76simpld 474 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)):𝐵1-1-onto→ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)))
8 f1ocnv 6187 . . . . . 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 3838 . . . . . 6 (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))) = (ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴)
116simprd 478 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥)) ∩ 𝐴) = ∅)
1210, 11syl5eq 2697 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))) = ∅)
132simprd 478 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∩ 𝐵) = ∅)
14 f1oun 6194 . . . . 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 1367 . . . 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 6165 . . . . 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 7687 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 +𝑜 𝐵) = (𝐴 ∪ ran (𝑥𝐵 ↦ (𝐴 +𝑜 𝑥))))
21 f1oeq2 6166 . . . 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 7687 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝐵 +𝑜 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))))
2524ancoms 468 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝐴) = (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))))
26 uncom 3790 . . . 4 (𝐵 ∪ ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥))) = (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵)
2725, 26syl6eq 2701 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 +𝑜 𝐴) = (ran (𝑥𝐴 ↦ (𝐵 +𝑜 𝑥)) ∪ 𝐵))
28 f1oeq3 6167 . . 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 383   = wceq 1523   ∈ wcel 2030   ∪ cun 3605   ∩ cin 3606  ∅c0 3948   ↦ cmpt 4762  ◡ccnv 5142  ran crn 5144  Oncon0 5761  –1-1-onto→wf1o 5925  (class class class)co 6690   +𝑜 coa 7602 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609 This theorem is referenced by:  cnfcomlem  8634
 Copyright terms: Public domain W3C validator