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Theorem fgreu 29811
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 6472 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹)
2 simplll 758 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Fun 𝐹)
3 funrel 6048 . . . . . . . 8 (Fun 𝐹 → Rel 𝐹)
42, 3syl 17 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Rel 𝐹)
5 simplr 752 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝𝐹)
6 1st2nd 7363 . . . . . . 7 ((Rel 𝐹𝑝𝐹) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
74, 5, 6syl2anc 573 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
8 simpr 471 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 = (1st𝑝))
9 simpllr 760 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 ∈ dom 𝐹)
108opeq1d 4545 . . . . . . . . . 10 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
117, 10eqtr4d 2808 . . . . . . . . 9 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (2nd𝑝)⟩)
1211, 5eqeltrrd 2851 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹)
13 funopfvb 6380 . . . . . . . . 9 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹𝑋) = (2nd𝑝) ↔ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹))
1413biimpar 463 . . . . . . . 8 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹) → (𝐹𝑋) = (2nd𝑝))
152, 9, 12, 14syl21anc 1475 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → (𝐹𝑋) = (2nd𝑝))
168, 15opeq12d 4547 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (𝐹𝑋)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
177, 16eqtr4d 2808 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
18 simpr 471 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
1918fveq2d 6336 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st𝑝) = (1st ‘⟨𝑋, (𝐹𝑋)⟩))
20 fvex 6342 . . . . . . . 8 (𝐹𝑋) ∈ V
21 op1stg 7327 . . . . . . . 8 ((𝑋 ∈ dom 𝐹 ∧ (𝐹𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2220, 21mpan2 671 . . . . . . 7 (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2322ad3antlr 710 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st ‘⟨𝑋, (𝐹𝑋)⟩) = 𝑋)
2419, 23eqtr2d 2806 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑋 = (1st𝑝))
2517, 24impbida 802 . . . 4 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) → (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2625ralrimiva 3115 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
27 eqeq2 2782 . . . . . 6 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (𝑝 = 𝑞𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2827bibi2d 331 . . . . 5 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → ((𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
2928ralbidv 3135 . . . 4 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
3029rspcev 3460 . . 3 ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 ∧ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
311, 26, 30syl2anc 573 . 2 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
32 reu6 3547 . 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 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  ∃!wreu 3063  Vcvv 3351  cop 4322  dom cdm 5249  Rel wrel 5254  Fun wfun 6025  cfv 6031  1st c1st 7313  2nd c2nd 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-1st 7315  df-2nd 7316
This theorem is referenced by:  fcnvgreu  29812
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