Theorem omf1o 8815

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 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
omf1o.2	⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))

Assertion

Ref	Expression
omf1o	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(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.

Step	Hyp	Ref	Expression
1		eqid 2738	. . . . . 6 ⊢ (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) = (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
2	1	omxpenlem 8813	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
3	2	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
4		eqid 2738	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})
5	4	xpcomf1o 8801	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6		f1oco 6722	. . . 4 ⊢ (((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)):(𝐴 × 𝐵)–1-1-onto→(𝐵 ·o 𝐴) ∧ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)) → ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴))
7	3, 5, 6	sylancl 585	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴))
8		omf1o.2	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
9	4, 1	xpcomco 8802	. . . . 5 ⊢ ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
10	8, 9	eqtr4i 2769	. . . 4 ⊢ 𝐺 = ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}))
11		f1oeq1 6688	. . . 4 ⊢ (𝐺 = ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})) → (𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴) ↔ ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴)))
12	10, 11	ax-mp 5	. . 3 ⊢ (𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴) ↔ ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})):(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴))
13	7, 12	sylibr 233	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴))
14		omf1o.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐴 ↦ ((𝐴 ·o 𝑥) +o 𝑦))
15	14	omxpenlem 8813	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
16		f1ocnv 6712	. . 3 ⊢ (𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵) → ◡𝐹:(𝐴 ·o 𝐵)–1-1-onto→(𝐵 × 𝐴))
17	15, 16	syl 17	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ◡𝐹:(𝐴 ·o 𝐵)–1-1-onto→(𝐵 × 𝐴))
18		f1oco 6722	. 2 ⊢ ((𝐺:(𝐵 × 𝐴)–1-1-onto→(𝐵 ·o 𝐴) ∧ ◡𝐹:(𝐴 ·o 𝐵)–1-1-onto→(𝐵 × 𝐴)) → (𝐺 ∘ ◡𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴))
19	13, 17, 18	syl2anc 583	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  {csn 4558  ∪ cuni 4836   ↦ cmpt 5153   × cxp 5578  ^◡ccnv 5579   ∘ ccom 5584  Oncon0 6251  –1-1-onto→wf1o 6417  (class class class)co 7255   ∈ cmpo 7257   +_o coa 8264   ·_o comu 8265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-omul 8272

This theorem is referenced by:  cnfcom3  9392  infxpenc  9705