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Theorem omf1o 8218
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 2770 . . . . . 6 (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) = (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥))
21omxpenlem 8216 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴))
32ancoms 455 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·𝑜 𝐴))
4 eqid 2770 . . . . 5 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})
54xpcomf1o 8204 . . . 4 (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6 f1oco 6300 . . . 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 566 . . 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 8205 . . . . 5 ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧})) = (𝑥𝐵, 𝑦𝐴 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥))
108, 9eqtr4i 2795 . . . 4 𝐺 = ((𝑦𝐴, 𝑥𝐵 ↦ ((𝐵 ·𝑜 𝑦) +𝑜 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ {𝑧}))
11 f1oeq1 6268 . . . 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 8216 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·𝑜 𝐵))
16 f1ocnv 6290 . . 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 6300 . 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 565 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 382   = wceq 1630  wcel 2144  {csn 4314   cuni 4572  cmpt 4861   × cxp 5247  ccnv 5248  ccom 5253  Oncon0 5866  1-1-ontowf1o 6030  (class class class)co 6792  cmpt2 6794   +𝑜 coa 7709   ·𝑜 comu 7710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-omul 7717
This theorem is referenced by:  cnfcom3  8764  infxpenc  9040
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