Theorem fgreu 32956

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 7046	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹)
2		simplll 786	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → Fun 𝐹)
3		funrel 6554	. . . . . . . 8 ⊢ (Fun 𝐹 → Rel 𝐹)
4	2, 3	syl 18	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → Rel 𝐹)
5		simplr 780	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 ∈ 𝐹)
6		1st2nd 8035	. . . . . . 7 ⊢ ((Rel 𝐹 ∧ 𝑝 ∈ 𝐹) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
7	4, 5, 6	syl2anc 595	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
8		simpr 489	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑋 = (1st ‘𝑝))
9		simpllr 787	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑋 ∈ dom 𝐹)
10	8	opeq1d 4848	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
11	7, 10	eqtr4d 2807	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = ⟨𝑋, (2nd ‘𝑝)⟩)
12	11, 5	eqeltrrd 2870	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹)
13		funopfvb 6936	. . . . . . . . 9 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ((𝐹‘𝑋) = (2nd ‘𝑝) ↔ ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹))
14	13	biimpar 482	. . . . . . . 8 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹) → (𝐹‘𝑋) = (2nd ‘𝑝))
15	2, 9, 12, 14	syl21anc 850	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → (𝐹‘𝑋) = (2nd ‘𝑝))
16	8, 15	opeq12d 4850	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (𝐹‘𝑋)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
17	7, 16	eqtr4d 2807	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)
18		simpr 489	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)
19	18	fveq2d 6886	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘𝑝) = (1st ‘⟨𝑋, (𝐹‘𝑋)⟩))
20		fvex 6895	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
21		op1stg 7997	. . . . . . . 8 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (𝐹‘𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
22	20, 21	mpan2 703	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
23	22	ad3antlr 743	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
24	19, 23	eqtr2d 2805	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → 𝑋 = (1st ‘𝑝))
25	17, 24	impbida 812	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) → (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
26	25	ralrimiva 3163	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
27		eqeq2 2781	. . . . . 6 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → (𝑝 = 𝑞 ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
28	27	bibi2d 345	. . . . 5 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → ((𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)))
29	28	ralbidv 3194	. . . 4 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → (∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)))
30	29	rspcev 3590	. . 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 595	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞))
32		reu6 3698	. 2 ⊢ (∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝) ↔ ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞))
33	31, 32	sylibr 237	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ∃!wreu 3374  Vcvv 3463  ⟨cop 4600  dom cdm 5662  Rel wrel 5667  Fun wfun 6531  ‘cfv 6537  1^st c1st 7983  2^nd c2nd 7984

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-1st 7985  df-2nd 7986

This theorem is referenced by:  fcnvgreu  32957