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

Theorem funcnvtp 6410
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 1128 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐴𝑈)
2 simp2 1129 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐶𝑉)
3 simp1 1128 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → 𝐵𝐷)
4 funcnvpr 6409 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
51, 2, 3, 4syl2an3an 1414 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
6 funcnvsn 6397 . . . 4 Fun {⟨𝐸, 𝐹⟩}
76a1i 11 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐸, 𝐹⟩})
8 df-rn 5559 . . . . . . 7 ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}
9 rnpropg 6072 . . . . . . 7 ((𝐴𝑈𝐶𝑉) → ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
108, 9syl5eqr 2867 . . . . . 6 ((𝐴𝑈𝐶𝑉) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
11103adant3 1124 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
12 df-rn 5559 . . . . . . 7 ran {⟨𝐸, 𝐹⟩} = dom {⟨𝐸, 𝐹⟩}
13 rnsnopg 6071 . . . . . . 7 (𝐸𝑊 → ran {⟨𝐸, 𝐹⟩} = {𝐹})
1412, 13syl5eqr 2867 . . . . . 6 (𝐸𝑊 → dom {⟨𝐸, 𝐹⟩} = {𝐹})
15143ad2ant3 1127 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐸, 𝐹⟩} = {𝐹})
1611, 15ineq12d 4187 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ({𝐵, 𝐷} ∩ {𝐹}))
17 disjprsn 4642 . . . . 5 ((𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
18173adant1 1122 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
1916, 18sylan9eq 2873 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ∅)
20 funun 6393 . . 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 833 . 2 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
22 df-tp 4562 . . . . 5 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2322cnveqi 5738 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
24 cnvun 5994 . . . 4 ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}) = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2523, 24eqtri 2841 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2625funeqi 6369 . 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 1079   = wceq 1528  wcel 2105  wne 3013  cun 3931  cin 3932  c0 4288  {csn 4557  {cpr 4559  {ctp 4561  cop 4563  ccnv 5547  dom cdm 5548  ran crn 5549  Fun wfun 6342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-fun 6350
This theorem is referenced by:  funcnvs3  14264
  Copyright terms: Public domain W3C validator