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Theorem fgreu 29456
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 6327 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹)
2 simplll 798 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Fun 𝐹)
3 funrel 5903 . . . . . . . 8 (Fun 𝐹 → Rel 𝐹)
42, 3syl 17 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → Rel 𝐹)
5 simplr 792 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝𝐹)
6 1st2nd 7211 . . . . . . 7 ((Rel 𝐹𝑝𝐹) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
74, 5, 6syl2anc 693 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
8 simpr 477 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 = (1st𝑝))
9 simpllr 799 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑋 ∈ dom 𝐹)
108opeq1d 4406 . . . . . . . . . 10 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
117, 10eqtr4d 2658 . . . . . . . . 9 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (2nd𝑝)⟩)
1211, 5eqeltrrd 2701 . . . . . . . 8 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹)
13 funopfvb 6237 . . . . . . . . 9 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ((𝐹𝑋) = (2nd𝑝) ↔ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹))
1413biimpar 502 . . . . . . . 8 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ ⟨𝑋, (2nd𝑝)⟩ ∈ 𝐹) → (𝐹𝑋) = (2nd𝑝))
152, 9, 12, 14syl21anc 1324 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → (𝐹𝑋) = (2nd𝑝))
168, 15opeq12d 4408 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → ⟨𝑋, (𝐹𝑋)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
177, 16eqtr4d 2658 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑋 = (1st𝑝)) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
18 simpr 477 . . . . . . 7 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)
1918fveq2d 6193 . . . . . 6 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → (1st𝑝) = (1st ‘⟨𝑋, (𝐹𝑋)⟩))
20 fvex 6199 . . . . . . . 8 (𝐹𝑋) ∈ V
21 op1stg 7177 . . . . . . . 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 2656 . . . . 5 ((((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩) → 𝑋 = (1st𝑝))
2517, 24impbida 877 . . . 4 (((Fun 𝐹𝑋 ∈ dom 𝐹) ∧ 𝑝𝐹) → (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2625ralrimiva 2965 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
27 eqeq2 2632 . . . . . 6 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (𝑝 = 𝑞𝑝 = ⟨𝑋, (𝐹𝑋)⟩))
2827bibi2d 332 . . . . 5 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → ((𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
2928ralbidv 2985 . . . 4 (𝑞 = ⟨𝑋, (𝐹𝑋)⟩ → (∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)))
3029rspcev 3307 . . 3 ((⟨𝑋, (𝐹𝑋)⟩ ∈ 𝐹 ∧ ∀𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹𝑋)⟩)) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
311, 26, 30syl2anc 693 . 2 ((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃𝑞𝐹𝑝𝐹 (𝑋 = (1st𝑝) ↔ 𝑝 = 𝑞))
32 reu6 3393 . 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 384   = wceq 1482  wcel 1989  wral 2911  wrex 2912  ∃!wreu 2913  Vcvv 3198  cop 4181  dom cdm 5112  Rel wrel 5117  Fun wfun 5880  cfv 5886  1st c1st 7163  2nd c2nd 7164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-fv 5894  df-1st 7165  df-2nd 7166
This theorem is referenced by:  fcnvgreu  29457
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