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Theorem fgreu 30333
Description: Exactly one point of a function's graph has a given first element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Assertion
Ref Expression
fgreu ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃!𝑝𝐹 𝑋 = (1st𝑝))
Distinct variable groups:   𝐹,𝑝   𝑋,𝑝

Proof of Theorem fgreu
Dummy variable 𝑞 is distinct from all other variables.
StepHypRef Expression
1 funfvop 6815 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹)
2 simplll 771 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Fun 𝐹)
3 funrel 6368 . . . . . . . 8 (Fun 𝐹 → Rel 𝐹)
42, 3syl 17 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Rel 𝐹)
5 simplr 765 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝𝐹)
6 1st2nd 7732 . . . . . . 7 ((Rel 𝐹𝑝𝐹) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
74, 5, 6syl2anc 584 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
8 simpr 485 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 = (1st𝑝))
9 simpllr 772 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 ∈ dom 𝐹)
108opeq1d 4807 . . . . . . . . . 10 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
117, 10eqtr4d 2863 . . . . . . . . 9 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (2nd𝑝)⟩)
1211, 5eqeltrrd 2918 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹)
13 funopfvb 6717 . . . . . . . . 9 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹𝑋) = (2nd𝑝) ↔ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹))
1413biimpar 478 . . . . . . . 8 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹) → (𝐹𝑋) = (2nd𝑝))
152, 9, 12, 14syl21anc 835 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → (𝐹𝑋) = (2nd𝑝))
168, 15opeq12d 4809 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (𝐹𝑋)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
177, 16eqtr4d 2863 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
18 simpr 485 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
1918fveq2d 6670 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st𝑝) = (1st ‘⟨𝑋, (𝐹𝑋)⟩))
20 fvex 6679 . . . . . . . 8 (𝐹𝑋) ∈ V
21 op1stg 7695 . . . . . . . 8 ((𝑋 ∈ dom 𝐹 ∧ (𝐹𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2220, 21mpan2 687 . . . . . . 7 (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2322ad3antlr 727 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2419, 23eqtr2d 2861 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑋 = (1st𝑝))
2517, 24impbida 797 . . . 4 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) → (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2625ralrimiva 3186 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
27 eqeq2 2836 . . . . . 6 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (𝑝 = 𝑞𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2827bibi2d 344 . . . . 5 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → ((𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
2928ralbidv 3201 . . . 4 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
3029rspcev 3626 . . 3 ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 ∧ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
311, 26, 30syl2anc 584 . 2 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
32 reu6 3720 . 2 (∃!𝑝𝐹 𝑋 = (1st𝑝) ↔ ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
3331, 32sylibr 235 1 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃!𝑝𝐹 𝑋 = (1st𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2106  wral 3142  wrex 3143  ∃!wreu 3144  Vcvv 3499  cop 4569  dom cdm 5553  Rel wrel 5558  Fun wfun 6345  cfv 6351  1st c1st 7681  2nd c2nd 7682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359  df-1st 7683  df-2nd 7684
This theorem is referenced by:  fcnvgreu  30334
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