Theorem omf1o 9018

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 2736	. . . . . 6 ⊢ (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) = (𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥))
2	1	omxpenlem 9016	. . . . 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 2736	. . . . 5 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}) = (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧})
5	4	xpcomf1o 9004	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}):(𝐵 × 𝐴)–1-1-onto→(𝐴 × 𝐵)
6		f1oco 6803	. . . 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 587	. . 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 9005	. . . . 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 2762	. . . 4 ⊢ 𝐺 = ((𝑦 ∈ 𝐴, 𝑥 ∈ 𝐵 ↦ ((𝐵 ·o 𝑦) +o 𝑥)) ∘ (𝑧 ∈ (𝐵 × 𝐴) ↦ ∪ ◡{𝑧}))
11		f1oeq1 6768	. . . 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 234	. 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 9016	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐹:(𝐵 × 𝐴)–1-1-onto→(𝐴 ·o 𝐵))
16		f1ocnv 6792	. . 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 6803	. 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 585	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐺 ∘ ◡𝐹):(𝐴 ·o 𝐵)–1-1-onto→(𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {csn 4567  ∪ cuni 4850   ↦ cmpt 5166   × cxp 5629  ^◡ccnv 5630   ∘ ccom 5635  Oncon0 6323  –1-1-onto→wf1o 6497  (class class class)co 7367   ∈ cmpo 7369   +_o coa 8402   ·_o comu 8403

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-omul 8410

This theorem is referenced by:  cnfcom3  9625  infxpenc  9940