Theorem funcnvpr 6630

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 6618	. . . 4 ⊢ Fun ◡{⟨𝐴, 𝐵⟩}
2		funcnvsn 6618	. . . 4 ⊢ Fun ◡{⟨𝐶, 𝐷⟩}
3	1, 2	pm3.2i 470	. . 3 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ∧ Fun ◡{⟨𝐶, 𝐷⟩})
4		df-rn 5700	. . . . . . 7 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
5		rnsnopg 6243	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
6	4, 5	eqtr3id 2789	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → dom ◡{⟨𝐴, 𝐵⟩} = {𝐵})
7		df-rn 5700	. . . . . . 7 ⊢ ran {⟨𝐶, 𝐷⟩} = dom ◡{⟨𝐶, 𝐷⟩}
8		rnsnopg 6243	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
9	7, 8	eqtr3id 2789	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom ◡{⟨𝐶, 𝐷⟩} = {𝐷})
10	6, 9	ineqan12d 4230	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
11	10	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
12		disjsn2 4717	. . . . 5 ⊢ (𝐵 ≠ 𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
13	12	3ad2ant3 1134	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
14	11, 13	eqtrd 2775	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ∅)
15		funun 6614	. . 3 ⊢ (((Fun ◡{⟨𝐴, 𝐵⟩} ∧ Fun ◡{⟨𝐶, 𝐷⟩}) ∧ (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ∅) → Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
16	3, 14, 15	sylancr 587	. 2 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
17		df-pr 4634	. . . . 5 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
18	17	cnveqi 5888	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
19		cnvun 6165	. . . 4 ⊢ ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
20	18, 19	eqtri 2763	. . 3 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
21	20	funeqi 6589	. 2 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ↔ Fun (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩}))
22	16, 21	sylibr 234	1 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → Fun ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938   ∪ cun 3961   ∩ cin 3962  ∅c0 4339  {csn 4631  {cpr 4633  ⟨cop 4637  ^◡ccnv 5688  dom cdm 5689  ran crn 5690  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-fun 6565

This theorem is referenced by:  funcnvtp  6631  funcnvqp  6632  funcnvs2  14949