Theorem funcnvpr 6609

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 6597	. . . 4 ⊢ Fun ◡{⟨𝐴, 𝐵⟩}
2		funcnvsn 6597	. . . 4 ⊢ Fun ◡{⟨𝐶, 𝐷⟩}
3	1, 2	pm3.2i 469	. . 3 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ∧ Fun ◡{⟨𝐶, 𝐷⟩})
4		df-rn 5686	. . . . . . 7 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
5		rnsnopg 6219	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
6	4, 5	eqtr3id 2784	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → dom ◡{⟨𝐴, 𝐵⟩} = {𝐵})
7		df-rn 5686	. . . . . . 7 ⊢ ran {⟨𝐶, 𝐷⟩} = dom ◡{⟨𝐶, 𝐷⟩}
8		rnsnopg 6219	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
9	7, 8	eqtr3id 2784	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom ◡{⟨𝐶, 𝐷⟩} = {𝐷})
10	6, 9	ineqan12d 4213	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
11	10	3adant3 1130	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
12		disjsn2 4715	. . . . 5 ⊢ (𝐵 ≠ 𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
13	12	3ad2ant3 1133	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
14	11, 13	eqtrd 2770	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ∅)
15		funun 6593	. . 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 4630	. . . . 5 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
18	17	cnveqi 5873	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
19		cnvun 6141	. . . 4 ⊢ ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
20	18, 19	eqtri 2758	. . 3 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
21	20	funeqi 6568	. 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 1085   = wceq 1539   ∈ wcel 2104   ≠ wne 2938   ∪ cun 3945   ∩ cin 3946  ∅c0 4321  {csn 4627  {cpr 4629  ⟨cop 4633  ^◡ccnv 5674  dom cdm 5675  ran crn 5676  Fun wfun 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6544

This theorem is referenced by:  funcnvtp  6610  funcnvqp  6611  funcnvs2  14868