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Theorem funcnvpr 5908
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 5894 . . . 4 Fun {⟨𝐴, 𝐵⟩}
2 funcnvsn 5894 . . . 4 Fun {⟨𝐶, 𝐷⟩}
31, 2pm3.2i 471 . . 3 (Fun {⟨𝐴, 𝐵⟩} ∧ Fun {⟨𝐶, 𝐷⟩})
4 df-rn 5085 . . . . . . 7 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐴, 𝐵⟩}
5 rnsnopg 5573 . . . . . . 7 (𝐴𝑈 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
64, 5syl5eqr 2669 . . . . . 6 (𝐴𝑈 → dom {⟨𝐴, 𝐵⟩} = {𝐵})
7 df-rn 5085 . . . . . . 7 ran {⟨𝐶, 𝐷⟩} = dom {⟨𝐶, 𝐷⟩}
8 rnsnopg 5573 . . . . . . 7 (𝐶𝑉 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
97, 8syl5eqr 2669 . . . . . 6 (𝐶𝑉 → dom {⟨𝐶, 𝐷⟩} = {𝐷})
106, 9ineqan12d 3794 . . . . 5 ((𝐴𝑈𝐶𝑉) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
11103adant3 1079 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
12 disjsn2 4217 . . . . 5 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
13123ad2ant3 1082 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
1411, 13eqtrd 2655 . . 3 ((𝐴𝑈𝐶𝑉𝐵𝐷) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ∅)
15 funun 5890 . . 3 (((Fun {⟨𝐴, 𝐵⟩} ∧ Fun {⟨𝐶, 𝐷⟩}) ∧ (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ∅) → Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
163, 14, 15sylancr 694 . 2 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
17 df-pr 4151 . . . . 5 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
1817cnveqi 5257 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
19 cnvun 5497 . . . 4 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2018, 19eqtri 2643 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2120funeqi 5868 . 2 (Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ↔ Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
2216, 21sylibr 224 1 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  cun 3553  cin 3554  c0 3891  {csn 4148  {cpr 4150  cop 4154  ccnv 5073  dom cdm 5074  ran crn 5075  Fun wfun 5841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-fun 5849
This theorem is referenced by:  funcnvtp  5909  funcnvqp  5910  funcnvqpOLD  5911  funcnvs2  13594
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