Theorem fgreu 32744

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 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝))

Distinct variable groups:   𝐹,𝑝   𝑋,𝑝

Proof of Theorem fgreu

Dummy variable 𝑞 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfvop 7002	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹)
2		simplll 775	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → Fun 𝐹)
3		funrel 6515	. . . . . . . 8 ⊢ (Fun 𝐹 → Rel 𝐹)
4	2, 3	syl 17	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → Rel 𝐹)
5		simplr 769	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 ∈ 𝐹)
6		1st2nd 7992	. . . . . . 7 ⊢ ((Rel 𝐹 ∧ 𝑝 ∈ 𝐹) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
7	4, 5, 6	syl2anc 585	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
8		simpr 484	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑋 = (1st ‘𝑝))
9		simpllr 776	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑋 ∈ dom 𝐹)
10	8	opeq1d 4822	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
11	7, 10	eqtr4d 2774	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = ⟨𝑋, (2nd ‘𝑝)⟩)
12	11, 5	eqeltrrd 2837	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹)
13		funopfvb 6894	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹‘𝑋) = (2nd ‘𝑝) ↔ ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹))
14	13	biimpar 477	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹) → (𝐹‘𝑋) = (2nd ‘𝑝))
15	2, 9, 12, 14	syl21anc 838	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → (𝐹‘𝑋) = (2nd ‘𝑝))
16	8, 15	opeq12d 4824	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (𝐹‘𝑋)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
17	7, 16	eqtr4d 2774	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)
18		simpr 484	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)
19	18	fveq2d 6844	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘𝑝) = (1st ‘⟨𝑋, (𝐹‘𝑋)⟩))
20		fvex 6853	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
21		op1stg 7954	. . . . . . . 8 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (𝐹‘𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
22	20, 21	mpan2 692	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
23	22	ad3antlr 732	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
24	19, 23	eqtr2d 2772	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → 𝑋 = (1st ‘𝑝))
25	17, 24	impbida 801	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) → (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
26	25	ralrimiva 3129	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
27		eqeq2 2748	. . . . . 6 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → (𝑝 = 𝑞 ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
28	27	bibi2d 342	. . . . 5 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → ((𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)))
29	28	ralbidv 3160	. . . 4 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → (∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)))
30	29	rspcev 3564	. . 3 ⊢ ((⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹 ∧ ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)) → ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞))
31	1, 26, 30	syl2anc 585	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞))
32		reu6 3672	. 2 ⊢ (∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝) ↔ ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞))
33	31, 32	sylibr 234	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  ∃!wreu 3340  Vcvv 3429  ⟨cop 4573  dom cdm 5631  Rel wrel 5636  Fun wfun 6492  ‘cfv 6498  1^st c1st 7940  2^nd c2nd 7941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6454  df-fun 6500  df-fn 6501  df-fv 6506  df-1st 7942  df-2nd 7943

This theorem is referenced by:  fcnvgreu  32745