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Theorem fcnvgreu 29333
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 5090 . . . 4 ran 𝐴 = dom 𝐴
21eleq2i 2690 . . 3 (𝑌 ∈ ran 𝐴𝑌 ∈ dom 𝐴)
3 fgreu 29332 . . . 4 ((Fun 𝐴𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
43adantll 749 . . 3 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ dom 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
52, 4sylan2b 492 . 2 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑞 𝐴𝑌 = (1st𝑞))
6 cnvcnvss 5553 . . . . . 6 𝐴𝐴
7 cnvssrndm 5621 . . . . . . . . . . 11 𝐴 ⊆ (ran 𝐴 × dom 𝐴)
87sseli 3583 . . . . . . . . . 10 (𝑞𝐴𝑞 ∈ (ran 𝐴 × dom 𝐴))
9 dfdm4 5281 . . . . . . . . . . 11 dom 𝐴 = ran 𝐴
101, 9xpeq12i 5102 . . . . . . . . . 10 (ran 𝐴 × dom 𝐴) = (dom 𝐴 × ran 𝐴)
118, 10syl6eleq 2708 . . . . . . . . 9 (𝑞𝐴𝑞 ∈ (dom 𝐴 × ran 𝐴))
12 2nd1st 7165 . . . . . . . . 9 (𝑞 ∈ (dom 𝐴 × ran 𝐴) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1311, 12syl 17 . . . . . . . 8 (𝑞𝐴 {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
1413eqcomd 2627 . . . . . . 7 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
15 relcnv 5467 . . . . . . . 8 Rel 𝐴
16 cnvf1olem 7227 . . . . . . . . 9 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → (⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩}))
1716simpld 475 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
1815, 17mpan 705 . . . . . . 7 ((𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞}) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
1914, 18mpdan 701 . . . . . 6 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
206, 19sseldi 3585 . . . . 5 (𝑞𝐴 → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
2120adantl 482 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑞𝐴) → ⟨(2nd𝑞), (1st𝑞)⟩ ∈ 𝐴)
22 simpll 789 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → Rel 𝐴)
23 simpr 477 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝𝐴)
24 relssdmrn 5620 . . . . . . . . . . 11 (Rel 𝐴𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2524adantr 481 . . . . . . . . . 10 ((Rel 𝐴 ∧ Fun 𝐴) → 𝐴 ⊆ (dom 𝐴 × ran 𝐴))
2625sselda 3587 . . . . . . . . 9 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 ∈ (dom 𝐴 × ran 𝐴))
27 2nd1st 7165 . . . . . . . . 9 (𝑝 ∈ (dom 𝐴 × ran 𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2826, 27syl 17 . . . . . . . 8 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
2928eqcomd 2627 . . . . . . 7 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})
30 cnvf1olem 7227 . . . . . . . 8 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → (⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩}))
3130simpld 475 . . . . . . 7 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → ⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴)
3222, 23, 29, 31syl12anc 1321 . . . . . 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 791 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞𝐴)
3514ad2antlr 762 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})
3616simprd 479 . . . . . . . . . 10 ((Rel 𝐴 ∧ (𝑞𝐴 ∧ ⟨(2nd𝑞), (1st𝑞)⟩ = {𝑞})) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
3733, 34, 35, 36syl12anc 1321 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = {⟨(2nd𝑞), (1st𝑞)⟩})
38 simpr 477 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
3938sneqd 4165 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4039cnveqd 5263 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4140unieqd 4417 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = {⟨(2nd𝑞), (1st𝑞)⟩})
4228ad2antrr 761 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → {𝑝} = ⟨(2nd𝑝), (1st𝑝)⟩)
4337, 41, 423eqtr2d 2661 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4430simprd 479 . . . . . . . . . . 11 ((Rel 𝐴 ∧ (𝑝𝐴 ∧ ⟨(2nd𝑝), (1st𝑝)⟩ = {𝑝})) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4522, 23, 29, 44syl12anc 1321 . . . . . . . . . 10 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
4645ad2antrr 761 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑝 = {⟨(2nd𝑝), (1st𝑝)⟩})
47 simpr 477 . . . . . . . . . . . 12 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)
4847sneqd 4165 . . . . . . . . . . 11 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
4948cnveqd 5263 . . . . . . . . . 10 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5049unieqd 4417 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = {⟨(2nd𝑝), (1st𝑝)⟩})
5113ad2antlr 762 . . . . . . . . 9 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → {𝑞} = ⟨(2nd𝑞), (1st𝑞)⟩)
5246, 50, 513eqtr2d 2661 . . . . . . . 8 (((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) ∧ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩) → 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩)
5343, 52impbida 876 . . . . . . 7 ((((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) ∧ 𝑞𝐴) → (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5453ralrimiva 2961 . . . . . 6 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
55 eqeq2 2632 . . . . . . . . 9 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (𝑞 = 𝑟𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩))
5655bibi2d 332 . . . . . . . 8 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → ((𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5756ralbidv 2981 . . . . . . 7 (𝑟 = ⟨(2nd𝑝), (1st𝑝)⟩ → (∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟) ↔ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)))
5857rspcev 3298 . . . . . 6 ((⟨(2nd𝑝), (1st𝑝)⟩ ∈ 𝐴 ∧ ∀𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = ⟨(2nd𝑝), (1st𝑝)⟩)) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
5932, 54, 58syl2anc 692 . . . . 5 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝𝐴) → ∃𝑟 𝐴𝑞 𝐴(𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ ↔ 𝑞 = 𝑟))
60 reu6 3381 . . . . 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 6163 . . . . . . 7 (2nd𝑞) ∈ V
63 fvex 6163 . . . . . . 7 (1st𝑞) ∈ V
6462, 63op2ndd 7131 . . . . . 6 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (2nd𝑝) = (1st𝑞))
6564eqeq2d 2631 . . . . 5 (𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩ → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6665adantl 482 . . . 4 (((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑝 = ⟨(2nd𝑞), (1st𝑞)⟩) → (𝑌 = (2nd𝑝) ↔ 𝑌 = (1st𝑞)))
6721, 61, 66reuxfr4d 29197 . . 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 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  ∃!wreu 2909  wss 3559  {csn 4153  cop 4159   cuni 4407   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  Rel wrel 5084  Fun wfun 5846  cfv 5852  1st c1st 7118  2nd c2nd 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860  df-1st 7120  df-2nd 7121
This theorem is referenced by:  gsummpt2co  29583
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