Theorem funcnvpr 6616

Description: The converse pair of ordered pairs is a function if the second members are different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.)

Assertion

Ref	Expression
funcnvpr	⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})

Proof of Theorem funcnvpr

Step	Hyp	Ref	Expression
1		funcnvsn 6604	. . . 4 ⊢ Fun ◡{⟨𝐴, 𝐵⟩}
2		funcnvsn 6604	. . . 4 ⊢ Fun ◡{⟨𝐶, 𝐷⟩}
3	1, 2	pm3.2i 469	. . 3 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ∧ Fun ◡{⟨𝐶, 𝐷⟩})
4		df-rn 5689	. . . . . . 7 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
5		rnsnopg 6227	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
6	4, 5	eqtr3id 2779	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → dom ◡{⟨𝐴, 𝐵⟩} = {𝐵})
7		df-rn 5689	. . . . . . 7 ⊢ ran {⟨𝐶, 𝐷⟩} = dom ◡{⟨𝐶, 𝐷⟩}
8		rnsnopg 6227	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
9	7, 8	eqtr3id 2779	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom ◡{⟨𝐶, 𝐷⟩} = {𝐷})
10	6, 9	ineqan12d 4212	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
11	10	3adant3 1129	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
12		disjsn2 4718	. . . . 5 ⊢ (𝐵 ≠ 𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
13	12	3ad2ant3 1132	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
14	11, 13	eqtrd 2765	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ∅)
15		funun 6600	. . 3 ⊢ (((Fun ◡{⟨𝐴, 𝐵⟩} ∧ Fun ◡{⟨𝐶, 𝐷⟩}) ∧ (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ∅) → Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
16	3, 14, 15	sylancr 585	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
17		df-pr 4633	. . . . 5 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
18	17	cnveqi 5877	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
19		cnvun 6149	. . . 4 ⊢ ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
20	18, 19	eqtri 2753	. . 3 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
21	20	funeqi 6575	. 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ↔ Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
22	16, 21	sylibr 233	1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ≠ wne 2929   ∪ cun 3942   ∩ cin 3943  ∅c0 4322  {csn 4630  {cpr 4632  ⟨cop 4636  ^◡ccnv 5677  dom cdm 5678  ran crn 5679  Fun wfun 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-mo 2528  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-fun 6551

This theorem is referenced by:  funcnvtp  6617  funcnvqp  6618  funcnvs2  14900