Theorem fgreu 32690

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 7083	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹)
2		simplll 774	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → Fun 𝐹)
3		funrel 6595	. . . . . . . 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 768	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 ∈ 𝐹)
6		1st2nd 8080	. . . . . . 7 ⊢ ((Rel 𝐹 ∧ 𝑝 ∈ 𝐹) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
7	4, 5, 6	syl2anc 583	. . . . . 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 775	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑋 ∈ dom 𝐹)
10	8	opeq1d 4903	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
11	7, 10	eqtr4d 2783	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = ⟨𝑋, (2nd ‘𝑝)⟩)
12	11, 5	eqeltrrd 2845	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹)
13		funopfvb 6976	. . . . . . . . 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 837	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → (𝐹‘𝑋) = (2nd ‘𝑝))
16	8, 15	opeq12d 4905	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (𝐹‘𝑋)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
17	7, 16	eqtr4d 2783	. . . . 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 6924	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘𝑝) = (1st ‘⟨𝑋, (𝐹‘𝑋)⟩))
20		fvex 6933	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
21		op1stg 8042	. . . . . . . 8 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (𝐹‘𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
22	20, 21	mpan2 690	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
23	22	ad3antlr 730	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
24	19, 23	eqtr2d 2781	. . . . 5 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → 𝑋 = (1st ‘𝑝))
25	17, 24	impbida 800	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) → (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
26	25	ralrimiva 3152	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
27		eqeq2 2752	. . . . . 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 3184	. . . 4 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → (∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)))
30	29	rspcev 3635	. . 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 583	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞))
32		reu6 3748	. 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 1537   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  ∃!wreu 3386  Vcvv 3488  ⟨cop 4654  dom cdm 5700  Rel wrel 5705  Fun wfun 6567  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-1st 8030  df-2nd 8031

This theorem is referenced by:  fcnvgreu  32691