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Theorem fgreu 31897
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 7052 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹)
2 simplll 774 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Fun 𝐹)
3 funrel 6566 . . . . . . . 8 (Fun 𝐹 → Rel 𝐹)
42, 3syl 17 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Rel 𝐹)
5 simplr 768 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝𝐹)
6 1st2nd 8025 . . . . . . 7 ((Rel 𝐹𝑝𝐹) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
74, 5, 6syl2anc 585 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
8 simpr 486 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 = (1st𝑝))
9 simpllr 775 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 ∈ dom 𝐹)
108opeq1d 4880 . . . . . . . . . 10 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
117, 10eqtr4d 2776 . . . . . . . . 9 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (2nd𝑝)⟩)
1211, 5eqeltrrd 2835 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹)
13 funopfvb 6948 . . . . . . . . 9 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹𝑋) = (2nd𝑝) ↔ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹))
1413biimpar 479 . . . . . . . 8 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹) → (𝐹𝑋) = (2nd𝑝))
152, 9, 12, 14syl21anc 837 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → (𝐹𝑋) = (2nd𝑝))
168, 15opeq12d 4882 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (𝐹𝑋)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
177, 16eqtr4d 2776 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
18 simpr 486 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
1918fveq2d 6896 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st𝑝) = (1st ‘⟨𝑋, (𝐹𝑋)⟩))
20 fvex 6905 . . . . . . . 8 (𝐹𝑋) ∈ V
21 op1stg 7987 . . . . . . . 8 ((𝑋 ∈ dom 𝐹 ∧ (𝐹𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2220, 21mpan2 690 . . . . . . 7 (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2322ad3antlr 730 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2419, 23eqtr2d 2774 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑋 = (1st𝑝))
2517, 24impbida 800 . . . 4 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) → (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2625ralrimiva 3147 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
27 eqeq2 2745 . . . . . 6 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (𝑝 = 𝑞𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2827bibi2d 343 . . . . 5 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → ((𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
2928ralbidv 3178 . . . 4 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
3029rspcev 3613 . . 3 ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 ∧ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
311, 26, 30syl2anc 585 . 2 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
32 reu6 3723 . 2 (∃!𝑝𝐹 𝑋 = (1st𝑝) ↔ ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
3331, 32sylibr 233 1 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃!𝑝𝐹 𝑋 = (1st𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  wrex 3071  ∃!wreu 3375  Vcvv 3475  cop 4635  dom cdm 5677  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-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:  fcnvgreu  31898
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