MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcnvpr Structured version   Visualization version   GIF version

Theorem funcnvpr 6196
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 6184 . . . 4 Fun {⟨𝐴, 𝐵⟩}
2 funcnvsn 6184 . . . 4 Fun {⟨𝐶, 𝐷⟩}
31, 2pm3.2i 464 . . 3 (Fun {⟨𝐴, 𝐵⟩} ∧ Fun {⟨𝐶, 𝐷⟩})
4 df-rn 5366 . . . . . . 7 ran {⟨𝐴, 𝐵⟩} = dom {⟨𝐴, 𝐵⟩}
5 rnsnopg 5868 . . . . . . 7 (𝐴𝑈 → ran {⟨𝐴, 𝐵⟩} = {𝐵})
64, 5syl5eqr 2828 . . . . . 6 (𝐴𝑈 → dom {⟨𝐴, 𝐵⟩} = {𝐵})
7 df-rn 5366 . . . . . . 7 ran {⟨𝐶, 𝐷⟩} = dom {⟨𝐶, 𝐷⟩}
8 rnsnopg 5868 . . . . . . 7 (𝐶𝑉 → ran {⟨𝐶, 𝐷⟩} = {𝐷})
97, 8syl5eqr 2828 . . . . . 6 (𝐶𝑉 → dom {⟨𝐶, 𝐷⟩} = {𝐷})
106, 9ineqan12d 4039 . . . . 5 ((𝐴𝑈𝐶𝑉) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
11103adant3 1123 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ({𝐵} ∩ {𝐷}))
12 disjsn2 4479 . . . . 5 (𝐵𝐷 → ({𝐵} ∩ {𝐷}) = ∅)
13123ad2ant3 1126 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → ({𝐵} ∩ {𝐷}) = ∅)
1411, 13eqtrd 2814 . . 3 ((𝐴𝑈𝐶𝑉𝐵𝐷) → (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ∅)
15 funun 6180 . . 3 (((Fun {⟨𝐴, 𝐵⟩} ∧ Fun {⟨𝐶, 𝐷⟩}) ∧ (dom {⟨𝐴, 𝐵⟩} ∩ dom {⟨𝐶, 𝐷⟩}) = ∅) → Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
163, 14, 15sylancr 581 . 2 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
17 df-pr 4401 . . . . 5 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
1817cnveqi 5542 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
19 cnvun 5792 . . . 4 ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2018, 19eqtri 2802 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2120funeqi 6156 . 2 (Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ↔ Fun ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}))
2216, 21sylibr 226 1 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  w3a 1071   = wceq 1601  wcel 2107  wne 2969  cun 3790  cin 3791  c0 4141  {csn 4398  {cpr 4400  cop 4404  ccnv 5354  dom cdm 5355  ran crn 5356  Fun wfun 6129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-fun 6137
This theorem is referenced by:  funcnvtp  6197  funcnvqp  6198  funcnvs2  14064
  Copyright terms: Public domain W3C validator