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

Theorem funcnvtp 6610
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 1134 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐴𝑈)
2 simp2 1135 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → 𝐶𝑉)
3 simp1 1134 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → 𝐵𝐷)
4 funcnvpr 6609 . . . 4 ((𝐴𝑈𝐶𝑉𝐵𝐷) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
51, 2, 3, 4syl2an3an 1420 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩})
6 funcnvsn 6597 . . . 4 Fun {⟨𝐸, 𝐹⟩}
76a1i 11 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐸, 𝐹⟩})
8 df-rn 5686 . . . . . . 7 ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩}
9 rnpropg 6220 . . . . . . 7 ((𝐴𝑈𝐶𝑉) → ran {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
108, 9eqtr3id 2784 . . . . . 6 ((𝐴𝑈𝐶𝑉) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
11103adant3 1130 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐵, 𝐷})
12 df-rn 5686 . . . . . . 7 ran {⟨𝐸, 𝐹⟩} = dom {⟨𝐸, 𝐹⟩}
13 rnsnopg 6219 . . . . . . 7 (𝐸𝑊 → ran {⟨𝐸, 𝐹⟩} = {𝐹})
1412, 13eqtr3id 2784 . . . . . 6 (𝐸𝑊 → dom {⟨𝐸, 𝐹⟩} = {𝐹})
15143ad2ant3 1133 . . . . 5 ((𝐴𝑈𝐶𝑉𝐸𝑊) → dom {⟨𝐸, 𝐹⟩} = {𝐹})
1611, 15ineq12d 4212 . . . 4 ((𝐴𝑈𝐶𝑉𝐸𝑊) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ({𝐵, 𝐷} ∩ {𝐹}))
17 disjprsn 4717 . . . . 5 ((𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
18173adant1 1128 . . . 4 ((𝐵𝐷𝐵𝐹𝐷𝐹) → ({𝐵, 𝐷} ∩ {𝐹}) = ∅)
1916, 18sylan9eq 2790 . . 3 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → (dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∩ dom {⟨𝐸, 𝐹⟩}) = ∅)
20 funun 6593 . . 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 4632 . . . . 5 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2322cnveqi 5873 . . . 4 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
24 cnvun 6141 . . . 4 ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}) = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2523, 24eqtri 2758 . . 3 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} = ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩})
2625funeqi 6568 . 2 (Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩} ↔ Fun ({⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} ∪ {⟨𝐸, 𝐹⟩}))
2721, 26sylibr 233 1 (((𝐴𝑈𝐶𝑉𝐸𝑊) ∧ (𝐵𝐷𝐵𝐹𝐷𝐹)) → Fun {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩, ⟨𝐸, 𝐹⟩})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1085   = wceq 1539  wcel 2104  wne 2938  cun 3945  cin 3946  c0 4321  {csn 4627  {cpr 4629  {ctp 4631  cop 4633  ccnv 5674  dom cdm 5675  ran crn 5676  Fun wfun 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-fun 6544
This theorem is referenced by:  funcnvs3  14869
  Copyright terms: Public domain W3C validator