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Theorem funcnvtp 6294
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 1129 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐴𝑈)
2 simp2 1130 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐶𝑉)
3 simp1 1129 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → 𝐵𝐷)
4 funcnvpr 6293 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
51, 2, 3, 4syl2an3an 1415 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
6 funcnvsn 6281 . . . 4 Fun {⟨𝐸, 𝐹⟩}
76a1i 11 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐸, 𝐹⟩})
8 df-rn 5461 . . . . . . 7 ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}
9 rnpropg 5961 . . . . . . 7 ((𝐴𝑈𝐶𝑉) → ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
108, 9syl5eqr 2847 . . . . . 6 ((𝐴𝑈𝐶𝑉) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
11103adant3 1125 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
12 df-rn 5461 . . . . . . 7 ran {⟨𝐸, 𝐹⟩} = dom {⟨𝐸, 𝐹⟩}
13 rnsnopg 5960 . . . . . . 7 (𝐸𝑊 → ran {⟨𝐸, 𝐹⟩} = {𝐹})
1412, 13syl5eqr 2847 . . . . . 6 (𝐸𝑊 → dom {⟨𝐸, 𝐹⟩} = {𝐹})
15143ad2ant3 1128 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐸, 𝐹⟩} = {𝐹})
1611, 15ineq12d 4116 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ({𝐵, 𝐷} ∩ {𝐹}))
17 disjprsn 4563 . . . . 5 ((𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
18173adant1 1123 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
1916, 18sylan9eq 2853 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ∅)
20 funun 6277 . . 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 834 . 2 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
22 df-tp 4483 . . . . 5 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2322cnveqi 5638 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
24 cnvun 5884 . . . 4 ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}) = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2523, 24eqtri 2821 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2625funeqi 6253 . 2 (Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} ↔ Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
2721, 26sylibr 235 1 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1080   = wceq 1525  wcel 2083  wne 2986  cun 3863  cin 3864  c0 4217  {csn 4478  {cpr 4480  {ctp 4482  cop 4484  ccnv 5449  dom cdm 5450  ran crn 5451  Fun wfun 6226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rab 3116  df-v 3442  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-fun 6234
This theorem is referenced by:  funcnvs3  14116
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