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Theorem omf1o 8608
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 2818 . . . . . 6 (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) = (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
21omxpenlem 8606 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
32ancoms 459 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
4 eqid 2818 . . . . 5 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})
54xpcomf1o 8594 . . . 4 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6 f1oco 6630 . . . 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 8595 . . . . 5 ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})) = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
108, 9eqtr4i 2844 . . . 4 𝐺 = ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}))
11 f1oeq1 6597 . . . 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 235 . 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 8606 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
16 f1ocnv 6620 . . 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 6630 . 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 207  wa 396   = wceq 1528  wcel 2105  {csn 4557   cuni 4830  cmpt 5137   × cxp 5546  ccnv 5547  ccom 5552  Oncon0 6184  1-1-ontowf1o 6347  (class class class)co 7145  cmpo 7147   +o coa 8088   ·o comu 8089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-omul 8096
This theorem is referenced by:  cnfcom3  9155  infxpenc  9432
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