Theorem funcnvpr 6551

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 6539	. . . 4 ⊢ Fun ◡{⟨𝐴, 𝐵⟩}
2		funcnvsn 6539	. . . 4 ⊢ Fun ◡{⟨𝐶, 𝐷⟩}
3	1, 2	pm3.2i 470	. . 3 ⊢ (Fun ◡{⟨𝐴, 𝐵⟩} ∧ Fun ◡{⟨𝐶, 𝐷⟩})
4		df-rn 5632	. . . . . . 7 ⊢ ran {⟨𝐴, 𝐵⟩} = dom ◡{⟨𝐴, 𝐵⟩}
5		rnsnopg 6176	. . . . . . 7 ⊢ (𝐴 ∈ 𝑈 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
6	4, 5	eqtr3id 2782	. . . . . 6 ⊢ (𝐴 ∈ 𝑈 → dom ◡{⟨𝐴, 𝐵⟩} = {𝐵})
7		df-rn 5632	. . . . . . 7 ⊢ ran {⟨𝐶, 𝐷⟩} = dom ◡{⟨𝐶, 𝐷⟩}
8		rnsnopg 6176	. . . . . . 7 ⊢ (𝐶 ∈ 𝑉 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
9	7, 8	eqtr3id 2782	. . . . . 6 ⊢ (𝐶 ∈ 𝑉 → dom ◡{⟨𝐶, 𝐷⟩} = {𝐷})
10	6, 9	ineqan12d 4171	. . . . 5 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
11	10	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
12		disjsn2 4666	. . . . 5 ⊢ (𝐵 ≠ 𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
13	12	3ad2ant3 1135	. . . 4 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
14	11, 13	eqtrd 2768	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐶 ∈ 𝑉 ∧ 𝐵 ≠ 𝐷) → (dom ◡{⟨𝐴, 𝐵⟩} ∩ dom ◡{⟨𝐶, 𝐷⟩}) = ∅)
15		funun 6535	. . 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 4580	. . . . 5 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
18	17	cnveqi 5820	. . . 4 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
19		cnvun 6097	. . . 4 ⊢ ◡({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
20	18, 19	eqtri 2756	. . 3 ⊢ ◡{⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (◡{⟨𝐴, 𝐵⟩} ∪ ◡{⟨𝐶, 𝐷⟩})
21	20	funeqi 6510	. 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 1541   ∈ wcel 2113   ≠ wne 2929   ∪ cun 3896   ∩ cin 3897  ∅c0 4282  {csn 4577  {cpr 4579  ⟨cop 4583  ^◡ccnv 5620  dom cdm 5621  ran crn 5622  Fun wfun 6483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-fun 6491

This theorem is referenced by:  funcnvtp  6552  funcnvqp  6553  funcnvs2  14827