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Theorem funcnvtp 5909
Description: The converse triple of ordered pairs is a function if the second members are pairwise different. Note that the second members need not be sets. (Contributed by AV, 23-Jan-2021.)
Assertion
Ref Expression
funcnvtp (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})

Proof of Theorem funcnvtp
StepHypRef Expression
1 simp1 1059 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐴𝑈)
2 simp2 1060 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐶𝑉)
3 simp1 1059 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → 𝐵𝐷)
4 funcnvpr 5908 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
51, 2, 3, 4syl2an3an 1383 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
6 funcnvsn 5894 . . . 4 Fun {⟨𝐸, 𝐹⟩}
76a1i 11 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐸, 𝐹⟩})
8 df-rn 5085 . . . . . . 7 ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}
9 rnpropg 5574 . . . . . . 7 ((𝐴𝑈𝐶𝑉) → ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
108, 9syl5eqr 2669 . . . . . 6 ((𝐴𝑈𝐶𝑉) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
11103adant3 1079 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
12 df-rn 5085 . . . . . . 7 ran {⟨𝐸, 𝐹⟩} = dom {⟨𝐸, 𝐹⟩}
13 rnsnopg 5573 . . . . . . 7 (𝐸𝑊 → ran {⟨𝐸, 𝐹⟩} = {𝐹})
1412, 13syl5eqr 2669 . . . . . 6 (𝐸𝑊 → dom {⟨𝐸, 𝐹⟩} = {𝐹})
15143ad2ant3 1082 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐸, 𝐹⟩} = {𝐹})
1611, 15ineq12d 3793 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ({𝐵, 𝐷} ∩ {𝐹}))
17 disjprsn 4220 . . . . 5 ((𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
18173adant1 1077 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
1916, 18sylan9eq 2675 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ∅)
20 funun 5890 . . 3 (((Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∧ Fun {⟨𝐸, 𝐹⟩}) ∧ (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ∅) → Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
215, 7, 19, 20syl21anc 1322 . 2 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
22 df-tp 4153 . . . . 5 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2322cnveqi 5257 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
24 cnvun 5497 . . . 4 ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}) = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2523, 24eqtri 2643 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2625funeqi 5868 . 2 (Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} ↔ Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
2721, 26sylibr 224 1 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
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  {ctp 4152  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-tp 4153  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:  funcnvs3  13595
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