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Theorem fgreu 29599
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 6369 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹)
2 simplll 813 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Fun 𝐹)
3 funrel 5943 . . . . . . . 8 (Fun 𝐹 → Rel 𝐹)
42, 3syl 17 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Rel 𝐹)
5 simplr 807 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝𝐹)
6 1st2nd 7258 . . . . . . 7 ((Rel 𝐹𝑝𝐹) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
74, 5, 6syl2anc 694 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
8 simpr 476 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 = (1st𝑝))
9 simpllr 815 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 ∈ dom 𝐹)
108opeq1d 4439 . . . . . . . . . 10 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
117, 10eqtr4d 2688 . . . . . . . . 9 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (2nd𝑝)⟩)
1211, 5eqeltrrd 2731 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹)
13 funopfvb 6277 . . . . . . . . 9 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹𝑋) = (2nd𝑝) ↔ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹))
1413biimpar 501 . . . . . . . 8 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹) → (𝐹𝑋) = (2nd𝑝))
152, 9, 12, 14syl21anc 1365 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → (𝐹𝑋) = (2nd𝑝))
168, 15opeq12d 4441 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (𝐹𝑋)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
177, 16eqtr4d 2688 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
18 simpr 476 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
1918fveq2d 6233 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st𝑝) = (1st ‘⟨𝑋, (𝐹𝑋)⟩))
20 fvex 6239 . . . . . . . 8 (𝐹𝑋) ∈ V
21 op1stg 7222 . . . . . . . 8 ((𝑋 ∈ dom 𝐹 ∧ (𝐹𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2220, 21mpan2 707 . . . . . . 7 (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2322ad3antlr 767 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2419, 23eqtr2d 2686 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑋 = (1st𝑝))
2517, 24impbida 895 . . . 4 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) → (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2625ralrimiva 2995 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
27 eqeq2 2662 . . . . . 6 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (𝑝 = 𝑞𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2827bibi2d 331 . . . . 5 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → ((𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
2928ralbidv 3015 . . . 4 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
3029rspcev 3340 . . 3 ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 ∧ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
311, 26, 30syl2anc 694 . 2 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
32 reu6 3428 . 2 (∃!𝑝𝐹 𝑋 = (1st𝑝) ↔ ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
3331, 32sylibr 224 1 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃!𝑝𝐹 𝑋 = (1st𝑝))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  ∃!wreu 2943  Vcvv 3231  cop 4216  dom cdm 5143  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-ral 2946  df-rex 2947  df-reu 2948  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:  fcnvgreu  29600
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