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Theorem fcnvgreu 31898
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 5688 . . . 4 ran 𝐴 = dom 𝐴
21eleq2i 2826 . . 3 (𝑌 ∈ ran 𝐴𝑌 ∈ dom 𝐴)
3 fgreu 31897 . . . 4 ((Fun 𝐴𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
43adantll 713 . . 3 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
52, 4sylan2b 595 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
6 cnvcnvss 6194 . . . . . 6 𝐴𝐴
7 cnvssrndm 6271 . . . . . . . . . . 11 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
87sseli 3979 . . . . . . . . . 10 (𝑞𝐴𝑞 ∈ (ran 𝐴 × dom 𝐴))
9 dfdm4 5896 . . . . . . . . . . 11 dom 𝐴 = ran 𝐴
101, 9xpeq12i 5705 . . . . . . . . . 10 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
118, 10eleqtrdi 2844 . . . . . . . . 9 (𝑞𝐴𝑞 ∈ (dom 𝐴 × ran 𝐴))
12 2nd1st 8024 . . . . . . . . 9 (𝑞 ∈ (dom 𝐴 × ran 𝐴) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1311, 12syl 17 . . . . . . . 8 (𝑞𝐴 {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1413eqcomd 2739 . . . . . . 7 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
15 relcnv 6104 . . . . . . . 8 Rel 𝐴
16 cnvf1olem 8096 . . . . . . . . 9 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → (⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩}))
1716simpld 496 . . . . . . . 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, 19sselid 3981 . . . . 5 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
2120adantl 483 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑞𝐴) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
22 simpll 766 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → Rel 𝐴)
23 simpr 486 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝𝐴)
24 relssdmrn 6268 . . . . . . . . . . 11 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2524adantr 482 . . . . . . . . . 10 ((Rel 𝐴 ∧ Fun 𝐴) → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2625sselda 3983 . . . . . . . . 9 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 ∈ (dom 𝐴 × ran 𝐴))
27 2nd1st 8024 . . . . . . . . 9 (𝑝 ∈ (dom 𝐴 × ran 𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2826, 27syl 17 . . . . . . . 8 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2928eqcomd 2739 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})
30 cnvf1olem 8096 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → (⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩}))
3130simpld 496 . . . . . . 7 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → ⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴)
3222, 23, 29, 31syl12anc 836 . . . . . 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 497 . . . . . . . . . 10 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
3733, 34, 35, 36syl12anc 836 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
38 simpr 486 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
3938sneqd 4641 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4039cnveqd 5876 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4140unieqd 4923 . . . . . . . . 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 2779 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4430simprd 497 . . . . . . . . . . 11 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4522, 23, 29, 44syl12anc 836 . . . . . . . . . 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 486 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4847sneqd 4641 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
4948cnveqd 5876 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5049unieqd 4923 . . . . . . . . 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 2779 . . . . . . . 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 3147 . . . . . 6 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
55 eqeq2 2745 . . . . . . . . 9 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (𝑞 = 𝑟𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5655bibi2d 343 . . . . . . . 8 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → ((𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5756ralbidv 3178 . . . . . . 7 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5857rspcev 3613 . . . . . 6 ((⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴 ∧ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
5932, 54, 58syl2anc 585 . . . . 5 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
60 reu6 3723 . . . . 5 (∃!𝑞 𝐴𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
6159, 60sylibr 233 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃!𝑞 𝐴𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
62 fvex 6905 . . . . . . 7 (2nd𝑞) ∈ V
63 fvex 6905 . . . . . . 7 (1st𝑞) ∈ V
6462, 63op2ndd 7986 . . . . . 6 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (2nd𝑝) = (1st𝑞))
6564eqeq2d 2744 . . . . 5 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6665adantl 483 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6721, 61, 66reuxfr1d 3747 . . 3 ((Rel 𝐴 ∧ Fun 𝐴) → (∃!𝑝𝐴 𝑌 = (2nd𝑝) ↔ ∃!𝑞 𝐴𝑌 = (1st𝑞)))
6867adantr 482 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → (∃!𝑝𝐴 𝑌 = (2nd𝑝) ↔ ∃!𝑞 𝐴𝑌 = (1st𝑞)))
695, 68mpbird 257 1 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝𝐴 𝑌 = (2nd𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375  wss 3949  {csn 4629  cop 4635   cuni 4909   × cxp 5675  ccnv 5676  dom cdm 5677  ran crn 5678  Rel wrel 5682  Fun wfun 6538  cfv 6544  1st c1st 7973  2nd c2nd 7974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-fv 6552  df-1st 7975  df-2nd 7976
This theorem is referenced by:  gsummpt2co  32200
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