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

Theorem funcnvtp 6402
 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 1133 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐴𝑈)
2 simp2 1134 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐶𝑉)
3 simp1 1133 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → 𝐵𝐷)
4 funcnvpr 6401 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
51, 2, 3, 4syl2an3an 1419 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
6 funcnvsn 6389 . . . 4 Fun {⟨𝐸, 𝐹⟩}
76a1i 11 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐸, 𝐹⟩})
8 df-rn 5538 . . . . . . 7 ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}
9 rnpropg 6055 . . . . . . 7 ((𝐴𝑈𝐶𝑉) → ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
108, 9syl5eqr 2807 . . . . . 6 ((𝐴𝑈𝐶𝑉) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
11103adant3 1129 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
12 df-rn 5538 . . . . . . 7 ran {⟨𝐸, 𝐹⟩} = dom {⟨𝐸, 𝐹⟩}
13 rnsnopg 6054 . . . . . . 7 (𝐸𝑊 → ran {⟨𝐸, 𝐹⟩} = {𝐹})
1412, 13syl5eqr 2807 . . . . . 6 (𝐸𝑊 → dom {⟨𝐸, 𝐹⟩} = {𝐹})
15143ad2ant3 1132 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐸, 𝐹⟩} = {𝐹})
1611, 15ineq12d 4120 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ({𝐵, 𝐷} ∩ {𝐹}))
17 disjprsn 4610 . . . . 5 ((𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
18173adant1 1127 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
1916, 18sylan9eq 2813 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ∅)
20 funun 6385 . . 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 836 . 2 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
22 df-tp 4530 . . . . 5 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2322cnveqi 5719 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
24 cnvun 5977 . . . 4 ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}) = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2523, 24eqtri 2781 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2625funeqi 6360 . 2 (Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} ↔ Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
2721, 26sylibr 237 1 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2951   ∪ cun 3858   ∩ cin 3859  ∅c0 4227  {csn 4525  {cpr 4527  {ctp 4529  ⟨cop 4531  ◡ccnv 5526  dom cdm 5527  ran crn 5528  Fun wfun 6333 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-br 5036  df-opab 5098  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-fun 6341 This theorem is referenced by:  funcnvs3  14328
 Copyright terms: Public domain W3C validator