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Theorem funcnvpr 6628
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 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})

Proof of Theorem funcnvpr
StepHypRef Expression
1 funcnvsn 6616 . . . 4 Fun {⟨𝐴, 𝐵⟩}
2 funcnvsn 6616 . . . 4 Fun {⟨𝐶, 𝐷⟩}
31, 2pm3.2i 470 . . 3 (Fun {⟨𝐴, 𝐵⟩} ∧ Fun {⟨𝐶, 𝐷⟩})
4 df-rn 5696 . . . . . . 7 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐴, 𝐵⟩}
5 rnsnopg 6241 . . . . . . 7 (𝐴𝑈 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
64, 5eqtr3id 2791 . . . . . 6 (𝐴𝑈 → dom {⟨𝐴, 𝐵⟩} = {𝐵})
7 df-rn 5696 . . . . . . 7 ran {⟨𝐶, 𝐷⟩} = dom {⟨𝐶, 𝐷⟩}
8 rnsnopg 6241 . . . . . . 7 (𝐶𝑉 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
97, 8eqtr3id 2791 . . . . . 6 (𝐶𝑉 → dom {⟨𝐶, 𝐷⟩} = {𝐷})
106, 9ineqan12d 4222 . . . . 5 ((𝐴𝑈𝐶𝑉) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
11103adant3 1133 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
12 disjsn2 4712 . . . . 5 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
13123ad2ant3 1136 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
1411, 13eqtrd 2777 . . 3 ((𝐴𝑈𝐶𝑉𝐵𝐷) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ∅)
15 funun 6612 . . 3 (((Fun {⟨𝐴, 𝐵⟩} ∧ Fun {⟨𝐶, 𝐷⟩}) ∧ (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ∅) → Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
163, 14, 15sylancr 587 . 2 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
17 df-pr 4629 . . . . 5 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
1817cnveqi 5885 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
19 cnvun 6162 . . . 4 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2018, 19eqtri 2765 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2120funeqi 6587 . 2 (Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ↔ Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
2216, 21sylibr 234 1 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  cun 3949  cin 3950  c0 4333  {csn 4626  {cpr 4628  cop 4632  ccnv 5684  dom cdm 5685  ran crn 5686  Fun wfun 6555
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-fun 6563
This theorem is referenced by:  funcnvtp  6629  funcnvqp  6630  funcnvs2  14952
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