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Theorem fcnvgreu 29600
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 5154 . . . 4 ran 𝐴 = dom 𝐴
21eleq2i 2722 . . 3 (𝑌 ∈ ran 𝐴𝑌 ∈ dom 𝐴)
3 fgreu 29599 . . . 4 ((Fun 𝐴𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
43adantll 750 . . 3 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
52, 4sylan2b 491 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
6 cnvcnvss 5624 . . . . . 6 𝐴𝐴
7 cnvssrndm 5695 . . . . . . . . . . 11 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
87sseli 3632 . . . . . . . . . 10 (𝑞𝐴𝑞 ∈ (ran 𝐴 × dom 𝐴))
9 dfdm4 5348 . . . . . . . . . . 11 dom 𝐴 = ran 𝐴
101, 9xpeq12i 5171 . . . . . . . . . 10 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
118, 10syl6eleq 2740 . . . . . . . . 9 (𝑞𝐴𝑞 ∈ (dom 𝐴 × ran 𝐴))
12 2nd1st 7257 . . . . . . . . 9 (𝑞 ∈ (dom 𝐴 × ran 𝐴) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1311, 12syl 17 . . . . . . . 8 (𝑞𝐴 {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1413eqcomd 2657 . . . . . . 7 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
15 relcnv 5538 . . . . . . . 8 Rel 𝐴
16 cnvf1olem 7320 . . . . . . . . 9 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → (⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩}))
1716simpld 474 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
1815, 17mpan 706 . . . . . . 7 ((𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞}) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
1914, 18mpdan 703 . . . . . 6 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
206, 19sseldi 3634 . . . . 5 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
2120adantl 481 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑞𝐴) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
22 simpll 805 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → Rel 𝐴)
23 simpr 476 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝𝐴)
24 relssdmrn 5694 . . . . . . . . . . 11 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2524adantr 480 . . . . . . . . . 10 ((Rel 𝐴 ∧ Fun 𝐴) → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2625sselda 3636 . . . . . . . . 9 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 ∈ (dom 𝐴 × ran 𝐴))
27 2nd1st 7257 . . . . . . . . 9 (𝑝 ∈ (dom 𝐴 × ran 𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2826, 27syl 17 . . . . . . . 8 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2928eqcomd 2657 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})
30 cnvf1olem 7320 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → (⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩}))
3130simpld 474 . . . . . . 7 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → ⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴)
3222, 23, 29, 31syl12anc 1364 . . . . . 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 807 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞𝐴)
3514ad2antlr 763 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
3616simprd 478 . . . . . . . . . 10 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
3733, 34, 35, 36syl12anc 1364 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
38 simpr 476 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
3938sneqd 4222 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4039cnveqd 5330 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4140unieqd 4478 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4228ad2antrr 762 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
4337, 41, 423eqtr2d 2691 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4430simprd 478 . . . . . . . . . . 11 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4522, 23, 29, 44syl12anc 1364 . . . . . . . . . 10 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4645ad2antrr 762 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
47 simpr 476 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4847sneqd 4222 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
4948cnveqd 5330 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5049unieqd 4478 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5113ad2antlr 763 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
5246, 50, 513eqtr2d 2691 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
5343, 52impbida 895 . . . . . . 7 ((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5453ralrimiva 2995 . . . . . 6 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
55 eqeq2 2662 . . . . . . . . 9 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (𝑞 = 𝑟𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5655bibi2d 331 . . . . . . . 8 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → ((𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5756ralbidv 3015 . . . . . . 7 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5857rspcev 3340 . . . . . 6 ((⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴 ∧ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
5932, 54, 58syl2anc 694 . . . . 5 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
60 reu6 3428 . . . . 5 (∃!𝑞 𝐴𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
6159, 60sylibr 224 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃!𝑞 𝐴𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
62 fvex 6239 . . . . . . 7 (2nd𝑞) ∈ V
63 fvex 6239 . . . . . . 7 (1st𝑞) ∈ V
6462, 63op2ndd 7221 . . . . . 6 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (2nd𝑝) = (1st𝑞))
6564eqeq2d 2661 . . . . 5 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6665adantl 481 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6721, 61, 66reuxfr4d 29457 . . 3 ((Rel 𝐴 ∧ Fun 𝐴) → (∃!𝑝𝐴 𝑌 = (2nd𝑝) ↔ ∃!𝑞 𝐴𝑌 = (1st𝑞)))
6867adantr 480 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → (∃!𝑝𝐴 𝑌 = (2nd𝑝) ↔ ∃!𝑞 𝐴𝑌 = (1st𝑞)))
695, 68mpbird 247 1 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝𝐴 𝑌 = (2nd𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  wss 3607  {csn 4210  cop 4216   cuni 4468   × cxp 5141  ccnv 5142  dom cdm 5143  ran crn 5144  Rel wrel 5148  Fun wfun 5920  cfv 5926  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-1st 7210  df-2nd 7211
This theorem is referenced by:  gsummpt2co  29908
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