Theorem fgreu 32329

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 7051	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹)
2		simplll 772	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → Fun 𝐹)
3		funrel 6565	. . . . . . . 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 766	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑝 ∈ 𝐹)
6		1st2nd 8029	. . . . . . 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 773	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → 𝑋 ∈ dom 𝐹)
10	8	opeq1d 4879	. . . . . . . . . 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 2833	. . . . . . . 8 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → ⟨𝑋, (2nd ‘𝑝)⟩ ∈ 𝐹)
13		funopfvb 6947	. . . . . . . . 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 835	. . . . . . 7 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑋 = (1st ‘𝑝)) → (𝐹‘𝑋) = (2nd ‘𝑝))
16	8, 15	opeq12d 4881	. . . . . 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 6895	. . . . . 6 ⊢ ((((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) ∧ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩) → (1st ‘𝑝) = (1st ‘⟨𝑋, (𝐹‘𝑋)⟩))
20		fvex 6904	. . . . . . . 8 ⊢ (𝐹‘𝑋) ∈ V
21		op1stg 7991	. . . . . . . 8 ⊢ ((𝑋 ∈ dom 𝐹 ∧ (𝐹‘𝑋) ∈ V) → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
22	20, 21	mpan2 688	. . . . . . 7 ⊢ (𝑋 ∈ dom 𝐹 → (1st ‘⟨𝑋, (𝐹‘𝑋)⟩) = 𝑋)
23	22	ad3antlr 728	. . . . . 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 798	. . . 4 ⊢ (((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) ∧ 𝑝 ∈ 𝐹) → (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
26	25	ralrimiva 3145	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩))
27		eqeq2 2743	. . . . . 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 3176	. . . 4 ⊢ (𝑞 = ⟨𝑋, (𝐹‘𝑋)⟩ → (∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞) ↔ ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = ⟨𝑋, (𝐹‘𝑋)⟩)))
30	29	rspcev 3612	. . 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 3722	. 2 ⊢ (∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝) ↔ ∃𝑞 ∈ 𝐹 ∀𝑝 ∈ 𝐹 (𝑋 = (1st ‘𝑝) ↔ 𝑝 = 𝑞))
33	31, 32	sylibr 233	1 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → ∃!𝑝 ∈ 𝐹 𝑋 = (1st ‘𝑝))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  ∃!wreu 3373  Vcvv 3473  ⟨cop 4634  dom cdm 5676  Rel wrel 5681  Fun wfun 6537  ‘cfv 6543  1^st c1st 7977  2^nd c2nd 7978

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551  df-1st 7979  df-2nd 7980

This theorem is referenced by:  fcnvgreu  32330