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Theorem omf1o 9116
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 2736 . . . . . 6 (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) = (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
21omxpenlem 9114 . . . . 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 2736 . . . . 5 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})
54xpcomf1o 9102 . . . 4 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6 f1oco 6870 . . . 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 9103 . . . . 5 ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})) = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
108, 9eqtr4i 2767 . . . 4 𝐺 = ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}))
11 f1oeq1 6835 . . . 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 9114 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
16 f1ocnv 6859 . . 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 6870 . 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 1539  wcel 2107  {csn 4625   cuni 4906  cmpt 5224   × cxp 5682  ccnv 5683  ccom 5688  Oncon0 6383  1-1-ontowf1o 6559  (class class class)co 7432  cmpo 7434   +o coa 8504   ·o comu 8505
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512
This theorem is referenced by:  cnfcom3  9745  infxpenc  10059
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