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Theorem fcnvgreu 30346
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 5559 . . . 4 ran 𝐴 = dom 𝐴
21eleq2i 2901 . . 3 (𝑌 ∈ ran 𝐴𝑌 ∈ dom 𝐴)
3 fgreu 30345 . . . 4 ((Fun 𝐴𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
43adantll 710 . . 3 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
52, 4sylan2b 593 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
6 cnvcnvss 6044 . . . . . 6 𝐴𝐴
7 cnvssrndm 6115 . . . . . . . . . . 11 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
87sseli 3960 . . . . . . . . . 10 (𝑞𝐴𝑞 ∈ (ran 𝐴 × dom 𝐴))
9 dfdm4 5757 . . . . . . . . . . 11 dom 𝐴 = ran 𝐴
101, 9xpeq12i 5576 . . . . . . . . . 10 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
118, 10eleqtrdi 2920 . . . . . . . . 9 (𝑞𝐴𝑞 ∈ (dom 𝐴 × ran 𝐴))
12 2nd1st 7726 . . . . . . . . 9 (𝑞 ∈ (dom 𝐴 × ran 𝐴) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1311, 12syl 17 . . . . . . . 8 (𝑞𝐴 {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1413eqcomd 2824 . . . . . . 7 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
15 relcnv 5960 . . . . . . . 8 Rel 𝐴
16 cnvf1olem 7794 . . . . . . . . 9 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → (⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩}))
1716simpld 495 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
1815, 17mpan 686 . . . . . . 7 ((𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞}) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
1914, 18mpdan 683 . . . . . 6 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
206, 19sseldi 3962 . . . . 5 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
2120adantl 482 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑞𝐴) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
22 simpll 763 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → Rel 𝐴)
23 simpr 485 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝𝐴)
24 relssdmrn 6114 . . . . . . . . . . 11 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2524adantr 481 . . . . . . . . . 10 ((Rel 𝐴 ∧ Fun 𝐴) → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2625sselda 3964 . . . . . . . . 9 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 ∈ (dom 𝐴 × ran 𝐴))
27 2nd1st 7726 . . . . . . . . 9 (𝑝 ∈ (dom 𝐴 × ran 𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2826, 27syl 17 . . . . . . . 8 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2928eqcomd 2824 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})
30 cnvf1olem 7794 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → (⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩}))
3130simpld 495 . . . . . . 7 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → ⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴)
3222, 23, 29, 31syl12anc 832 . . . . . 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 765 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞𝐴)
3514ad2antlr 723 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
3616simprd 496 . . . . . . . . . 10 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
3733, 34, 35, 36syl12anc 832 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
38 simpr 485 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
3938sneqd 4569 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4039cnveqd 5739 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4140unieqd 4840 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4228ad2antrr 722 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
4337, 41, 423eqtr2d 2859 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4430simprd 496 . . . . . . . . . . 11 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4522, 23, 29, 44syl12anc 832 . . . . . . . . . 10 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4645ad2antrr 722 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
47 simpr 485 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4847sneqd 4569 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
4948cnveqd 5739 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5049unieqd 4840 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5113ad2antlr 723 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
5246, 50, 513eqtr2d 2859 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
5343, 52impbida 797 . . . . . . 7 ((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5453ralrimiva 3179 . . . . . 6 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
55 eqeq2 2830 . . . . . . . . 9 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (𝑞 = 𝑟𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5655bibi2d 344 . . . . . . . 8 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → ((𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5756ralbidv 3194 . . . . . . 7 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5857rspcev 3620 . . . . . 6 ((⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴 ∧ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
5932, 54, 58syl2anc 584 . . . . 5 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
60 reu6 3714 . . . . 5 (∃!𝑞 𝐴𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
6159, 60sylibr 235 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃!𝑞 𝐴𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
62 fvex 6676 . . . . . . 7 (2nd𝑞) ∈ V
63 fvex 6676 . . . . . . 7 (1st𝑞) ∈ V
6462, 63op2ndd 7689 . . . . . 6 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (2nd𝑝) = (1st𝑞))
6564eqeq2d 2829 . . . . 5 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6665adantl 482 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6721, 61, 66reuxfr1d 3738 . . 3 ((Rel 𝐴 ∧ Fun 𝐴) → (∃!𝑝𝐴 𝑌 = (2nd𝑝) ↔ ∃!𝑞 𝐴𝑌 = (1st𝑞)))
6867adantr 481 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → (∃!𝑝𝐴 𝑌 = (2nd𝑝) ↔ ∃!𝑞 𝐴𝑌 = (1st𝑞)))
695, 68mpbird 258 1 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝𝐴 𝑌 = (2nd𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  ∃!wreu 3137  wss 3933  {csn 4557  cop 4563   cuni 4830   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  Rel wrel 5553  Fun wfun 6342  cfv 6348  1st c1st 7676  2nd c2nd 7677
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-pow 5257  ax-pr 5320  ax-un 7450
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-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  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-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356  df-1st 7678  df-2nd 7679
This theorem is referenced by:  gsummpt2co  30613
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