Theorem fvrnressn 7152

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 5680	. . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
2	1	eleq2i 2817	. 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3		opeq1 4866	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨𝑥, (𝐹‘𝑋)⟩ = ⟨𝑋, (𝐹‘𝑋)⟩)
4	3	eleq1d 2810	. . . 4 ⊢ (𝑥 = 𝑋 → (⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
5	4	spcegv 3579	. . 3 ⊢ (𝑋 ∈ 𝑉 → (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹 → ∃𝑥⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹))
6		fvex 6895	. . . 4 ⊢ (𝐹‘𝑋) ∈ V
7		elimasng 6078	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐹‘𝑋) ∈ V) → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
8	6, 7	mpan2 688	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
9		elrn2g 5881	. . . 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 294	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
12	2, 11	biimtrrid 242	1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3466  {csn 4621  ⟨cop 4627  ran crn 5668   ↾ cres 5669   “ cima 5670  ‘cfv 6534

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-cnv 5675  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fv 6542

This theorem is referenced by:  fvn0fvelrnOLD  7154  funressndmfvrn  46264