Theorem funcnvpr 6547

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 6535	. . . 4 ⊢ Fun ◡{⟨𝐴, 𝐵⟩}
2		funcnvsn 6535	. . . 4 ⊢ Fun ◡{⟨𝐶, 𝐷⟩}
3	1, 2	pm3.2i 471	. . 3 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ∧ Fun ◡{⟨𝐶, 𝐷⟩})
4		df-rn 5629	. . . . . . 7 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
5		rnsnopg 6172	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
6	4, 5	eqtr3id 2788	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → dom ◡{⟨𝐴, 𝐵⟩} = {𝐵})
7		df-rn 5629	. . . . . . 7 ⊢ ran {⟨𝐶, 𝐷⟩} = dom ◡{⟨𝐶, 𝐷⟩}
8		rnsnopg 6172	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
9	7, 8	eqtr3id 2788	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom ◡{⟨𝐶, 𝐷⟩} = {𝐷})
10	6, 9	ineqan12d 4151	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
11	10	3adant3 1138	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
12		disjsn2 4644	. . . . 5 ⊢ (𝐵 ≠ 𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
13	12	3ad2ant3 1141	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
14	11, 13	eqtrd 2774	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ∅)
15		funun 6531	. . 3 ⊢ (((Fun ◡{⟨𝐴, 𝐵⟩} ∧ Fun ◡{⟨𝐶, 𝐷⟩}) ∧ (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ∅) → Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
16	3, 14, 15	sylancr 593	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
17		df-pr 4558	. . . . 5 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
18	17	cnveqi 5816	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
19		cnvun 6093	. . . 4 ⊢ ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
20	18, 19	eqtri 2762	. . 3 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
21	20	funeqi 6506	. 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ↔ Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
22	16, 21	sylibr 235	1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   ∪ cun 3881   ∩ cin 3882  ∅c0 4261  {csn 4555  {cpr 4557  ⟨cop 4561  ^◡ccnv 5617  dom cdm 5618  ran crn 5619  Fun wfun 6479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-fun 6487

This theorem is referenced by:  funcnvtp  6548  funcnvqp  6549  funcnvs2  14866