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Theorem omf1o 8023
Description: Construct an explicit bijection from 𝐴 ·𝑜 𝐵 to 𝐵 ·𝑜 𝐴. (Contributed by Mario Carneiro, 30-May-2015.)
Hypotheses
Ref Expression
omf1o.1 𝐹 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
omf1o.2 𝐺 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥))
Assertion
Ref Expression
omf1o ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺𝐹):(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)

Proof of Theorem omf1o
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . . 6 (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) = (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥))
21omxpenlem 8021 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴))
32ancoms 469 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴))
4 eqid 2621 . . . . 5 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})
54xpcomf1o 8009 . . . 4 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6 f1oco 6126 . . . 4 (((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴) ∧ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)) → ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·𝑜 𝐴))
73, 5, 6sylancl 693 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·𝑜 𝐴))
8 omf1o.2 . . . . 5 𝐺 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥))
94, 1xpcomco 8010 . . . . 5 ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})) = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥))
108, 9eqtr4i 2646 . . . 4 𝐺 = ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}))
11 f1oeq1 6094 . . . 4 (𝐺 = ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})) → (𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·𝑜 𝐴) ↔ ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·𝑜 𝐴)))
1210, 11ax-mp 5 . . 3 (𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·𝑜 𝐴) ↔ ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·𝑜 𝐴))
137, 12sylibr 224 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·𝑜 𝐴))
14 omf1o.1 . . . 4 𝐹 = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐴 ·𝑜 𝑥) +𝑜 𝑦))
1514omxpenlem 8021 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵))
16 f1ocnv 6116 . . 3 (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵) → 𝐹:(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 × 𝐴))
1715, 16syl 17 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 × 𝐴))
18 f1oco 6126 . 2 ((𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·𝑜 𝐴) ∧ 𝐹:(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 × 𝐴)) → (𝐺𝐹):(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴))
1913, 17, 18syl2anc 692 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺𝐹):(𝐴 ·𝑜 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {csn 4155   cuni 4409  cmpt 4683   × cxp 5082  ccnv 5083  ccom 5088  Oncon0 5692  1-1-ontowf1o 5856  (class class class)co 6615  cmpt2 6617   +𝑜 coa 7517   ·𝑜 comu 7518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-omul 7525
This theorem is referenced by:  cnfcom3  8561  infxpenc  8801
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