Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fcnvgreu Structured version   Visualization version   GIF version

Theorem fcnvgreu 30534
 Description: If the converse of a relation 𝐴 is a function, exactly one point of its graph has a given second element (that is, function value). (Contributed by Thierry Arnoux, 1-Apr-2018.)
Assertion
Ref Expression
fcnvgreu (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝𝐴 𝑌 = (2nd𝑝))
Distinct variable groups:   𝐴,𝑝   𝑌,𝑝

Proof of Theorem fcnvgreu
Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rn 5535 . . . 4 ran 𝐴 = dom 𝐴
21eleq2i 2843 . . 3 (𝑌 ∈ ran 𝐴𝑌 ∈ dom 𝐴)
3 fgreu 30533 . . . 4 ((Fun 𝐴𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
43adantll 713 . . 3 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
52, 4sylan2b 596 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
6 cnvcnvss 6023 . . . . . 6 𝐴𝐴
7 cnvssrndm 6100 . . . . . . . . . . 11 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
87sseli 3888 . . . . . . . . . 10 (𝑞𝐴𝑞 ∈ (ran 𝐴 × dom 𝐴))
9 dfdm4 5735 . . . . . . . . . . 11 dom 𝐴 = ran 𝐴
101, 9xpeq12i 5552 . . . . . . . . . 10 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
118, 10eleqtrdi 2862 . . . . . . . . 9 (𝑞𝐴𝑞 ∈ (dom 𝐴 × ran 𝐴))
12 2nd1st 7741 . . . . . . . . 9 (𝑞 ∈ (dom 𝐴 × ran 𝐴) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1311, 12syl 17 . . . . . . . 8 (𝑞𝐴 {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1413eqcomd 2764 . . . . . . 7 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
15 relcnv 5939 . . . . . . . 8 Rel 𝐴
16 cnvf1olem 7810 . . . . . . . . 9 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → (⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩}))
1716simpld 498 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
1815, 17mpan 689 . . . . . . 7 ((𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞}) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
1914, 18mpdan 686 . . . . . 6 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
206, 19sseldi 3890 . . . . 5 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
2120adantl 485 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑞𝐴) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
22 simpll 766 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → Rel 𝐴)
23 simpr 488 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝𝐴)
24 relssdmrn 6098 . . . . . . . . . . 11 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2524adantr 484 . . . . . . . . . 10 ((Rel 𝐴 ∧ Fun 𝐴) → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2625sselda 3892 . . . . . . . . 9 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 ∈ (dom 𝐴 × ran 𝐴))
27 2nd1st 7741 . . . . . . . . 9 (𝑝 ∈ (dom 𝐴 × ran 𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2826, 27syl 17 . . . . . . . 8 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2928eqcomd 2764 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})
30 cnvf1olem 7810 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → (⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩}))
3130simpld 498 . . . . . . 7 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → ⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴)
3222, 23, 29, 31syl12anc 835 . . . . . 6 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴)
3315a1i 11 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → Rel 𝐴)
34 simplr 768 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞𝐴)
3514ad2antlr 726 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
3616simprd 499 . . . . . . . . . 10 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
3733, 34, 35, 36syl12anc 835 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
38 simpr 488 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
3938sneqd 4534 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4039cnveqd 5715 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4140unieqd 4812 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4228ad2antrr 725 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
4337, 41, 423eqtr2d 2799 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4430simprd 499 . . . . . . . . . . 11 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4522, 23, 29, 44syl12anc 835 . . . . . . . . . 10 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4645ad2antrr 725 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
47 simpr 488 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4847sneqd 4534 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
4948cnveqd 5715 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5049unieqd 4812 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5113ad2antlr 726 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
5246, 50, 513eqtr2d 2799 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
5343, 52impbida 800 . . . . . . 7 ((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5453ralrimiva 3113 . . . . . 6 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
55 eqeq2 2770 . . . . . . . . 9 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (𝑞 = 𝑟𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5655bibi2d 346 . . . . . . . 8 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → ((𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5756ralbidv 3126 . . . . . . 7 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5857rspcev 3541 . . . . . 6 ((⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴 ∧ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
5932, 54, 58syl2anc 587 . . . . 5 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
60 reu6 3640 . . . . 5 (∃!𝑞 𝐴𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
6159, 60sylibr 237 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃!𝑞 𝐴𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
62 fvex 6671 . . . . . . 7 (2nd𝑞) ∈ V
63 fvex 6671 . . . . . . 7 (1st𝑞) ∈ V
6462, 63op2ndd 7704 . . . . . 6 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (2nd𝑝) = (1st𝑞))
6564eqeq2d 2769 . . . . 5 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6665adantl 485 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6721, 61, 66reuxfr1d 3664 . . 3 ((Rel 𝐴 ∧ Fun 𝐴) → (∃!𝑝𝐴 𝑌 = (2nd𝑝) ↔ ∃!𝑞 𝐴𝑌 = (1st𝑞)))
6867adantr 484 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → (∃!𝑝𝐴 𝑌 = (2nd𝑝) ↔ ∃!𝑞 𝐴𝑌 = (1st𝑞)))
695, 68mpbird 260 1 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝𝐴 𝑌 = (2nd𝑝))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  ∃!wreu 3072   ⊆ wss 3858  {csn 4522  ⟨cop 4528  ∪ cuni 4798   × cxp 5522  ◡ccnv 5523  dom cdm 5524  ran crn 5525  Rel wrel 5529  Fun wfun 6329  ‘cfv 6335  1st c1st 7691  2nd c2nd 7692 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 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 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-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-iota 6294  df-fun 6337  df-fn 6338  df-fv 6343  df-1st 7693  df-2nd 7694 This theorem is referenced by:  gsummpt2co  30834
 Copyright terms: Public domain W3C validator