Theorem fvrnressn 7133

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 5651	. . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
2	1	eleq2i 2820	. 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3		opeq1 4837	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨𝑥, (𝐹‘𝑋)⟩ = ⟨𝑋, (𝐹‘𝑋)⟩)
4	3	eleq1d 2813	. . . 4 ⊢ (𝑥 = 𝑋 → (⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
5	4	spcegv 3563	. . 3 ⊢ (𝑋 ∈ 𝑉 → (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹 → ∃𝑥⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹))
6		fvex 6871	. . . 4 ⊢ (𝐹‘𝑋) ∈ V
7		elimasng 6060	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐹‘𝑋) ∈ V) → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
8	6, 7	mpan2 691	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
9		elrn2g 5854	. . . 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 243	1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3447  {csn 4589  ⟨cop 4595  ran crn 5639   ↾ cres 5640   “ cima 5641  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fv 6519

This theorem is referenced by:  fvn0fvelrnOLD  7135  funressndmfvrn  47045