Theorem fvrnressn 6393

Description: If the value of a function is in the range of the function restricted to the singleton containing the argument, then the value of the function is in the range of the function. (Contributed by Alexander van der Vekens, 22-Jul-2018.)

Assertion

Ref	Expression
fvrnressn	⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))

Proof of Theorem fvrnressn

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ima 5097	. . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
2	1	eleq2i 2690	. 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3		opeq1 4377	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨𝑥, (𝐹‘𝑋)⟩ = ⟨𝑋, (𝐹‘𝑋)⟩)
4	3	eleq1d 2683	. . . 4 ⊢ (𝑥 = 𝑋 → (⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
5	4	spcegv 3284	. . 3 ⊢ (𝑋 ∈ 𝑉 → (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹 → ∃𝑥⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹))
6		fvex 6168	. . . 4 ⊢ (𝐹‘𝑋) ∈ V
7		elimasng 5460	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐹‘𝑋) ∈ V) → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
8	6, 7	mpan2 706	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
9		elrn2g 5283	. . . 4 ⊢ ((𝐹‘𝑋) ∈ V → ((𝐹‘𝑋) ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹))
10	6, 9	mp1i 13	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran 𝐹 ↔ ∃𝑥⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹))
11	5, 8, 10	3imtr4d 283	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
12	2, 11	syl5bir 233	1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1480  ∃wex 1701   ∈ wcel 1987  Vcvv 3190  {csn 4155  ⟨cop 4161  ran crn 5085   ↾ cres 5086   “ cima 5087  ‘cfv 5857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-cnv 5092  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fv 5865

This theorem is referenced by:  fvn0fvelrn  6395