Theorem fgreu 32699

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 6993	. . 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 6507	. . . . . . . 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 7981	. . . . . . 7 ⊢ ((Rel 𝐹 ∧ 𝑝 ∈ 𝐹) → 𝑝 = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
7	4, 5, 6	syl2anc 584	. . . . . 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 4833	. . . . . . . . . 10 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
11	7, 10	eqtr4d 2772	. . . . . . . . 9 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 = ⟨𝑋, (2nd ‘𝑝)⟩)
12	11, 5	eqeltrrd 2835	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹)
13		funopfvb 6886	. . . . . . . . 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 4835	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (𝐹‘𝑋)⟩ = ⟨(1st ‘𝑝), (2nd ‘𝑝)⟩)
17	7, 16	eqtr4d 2772	. . . . 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 6836	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘𝑝) = (1st ‘⟨𝑋, (𝐹‘𝑋)⟩))
20		fvex 6845	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
21		op1stg 7943	. . . . . . . 8 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (𝐹‘𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
22	20, 21	mpan2 691	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
23	22	ad3antlr 731	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
24	19, 23	eqtr2d 2770	. . . . 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 3126	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
27		eqeq2 2746	. . . . . 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 3157	. . . 4 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → (∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)))
30	29	rspcev 3574	. . 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 584	. 2 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞))
32		reu6 3682	. 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 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  ∃!wreu 3346  Vcvv 3438  ⟨cop 4584  dom cdm 5622  Rel wrel 5627  Fun wfun 6484  ‘cfv 6490  1^st c1st 7929  2^nd c2nd 7930

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498  df-1st 7931  df-2nd 7932

This theorem is referenced by:  fcnvgreu  32700