Theorem fvrnressn 7146

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 5662	. . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
2	1	eleq2i 2856	. 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3		opeq1 4833	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨𝑥, (𝐹‘𝑋)⟩ = ⟨𝑋, (𝐹‘𝑋)⟩)
4	3	eleq1d 2849	. . . 4 ⊢ (𝑥 = 𝑋 → (⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
5	4	spcegv 3558	. . 3 ⊢ (𝑋 ∈ 𝑉 → (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹 → ∃𝑥⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹))
6		fvex 6882	. . . 4 ⊢ (𝐹‘𝑋) ∈ V
7		elimasng 6080	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐹‘𝑋) ∈ V) → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
8	6, 7	mpan2 701	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
9		elrn2g 5868	. . . 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 296	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
12	2, 11	biimtrrid 245	1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1562  ∃wex 1801   ∈ wcel 2144  Vcvv 3456  {csn 4584  ⟨cop 4590  ran crn 5650   ↾ cres 5651   “ cima 5652  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5655  df-cnv 5657  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fv 6531

This theorem is referenced by:  funressndmfvrn  47643