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Theorem omf1o 9114
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 2735 . . . . . 6 (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) = (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
21omxpenlem 9112 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
32ancoms 458 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
4 eqid 2735 . . . . 5 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})
54xpcomf1o 9100 . . . 4 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6 f1oco 6872 . . . 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 586 . . 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 9101 . . . . 5 ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})) = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
108, 9eqtr4i 2766 . . . 4 𝐺 = ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}))
11 f1oeq1 6837 . . . 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 234 . 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 9112 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
16 f1ocnv 6861 . . 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 6872 . 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 584 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 206  wa 395   = wceq 1537  wcel 2106  {csn 4631   cuni 4912  cmpt 5231   × cxp 5687  ccnv 5688  ccom 5693  Oncon0 6386  1-1-ontowf1o 6562  (class class class)co 7431  cmpo 7433   +o coa 8502   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510
This theorem is referenced by:  cnfcom3  9742  infxpenc  10056
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