Theorem fvrnressn 6900

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 5532	. . 3 ⊢ (𝐹 “ {𝑋}) = ran (𝐹 ↾ {𝑋})
2	1	eleq2i 2881	. 2 ⊢ ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ (𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}))
3		opeq1 4763	. . . . 5 ⊢ (𝑥 = 𝑋 → ⟨𝑥, (𝐹‘𝑋)⟩ = ⟨𝑋, (𝐹‘𝑋)⟩)
4	3	eleq1d 2874	. . . 4 ⊢ (𝑥 = 𝑋 → (⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹 ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
5	4	spcegv 3545	. . 3 ⊢ (𝑋 ∈ 𝑉 → (⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹 → ∃𝑥⟨𝑥, (𝐹‘𝑋)⟩ ∈ 𝐹))
6		fvex 6658	. . . 4 ⊢ (𝐹‘𝑋) ∈ V
7		elimasng 5922	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝐹‘𝑋) ∈ V) → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
8	6, 7	mpan2 690	. . 3 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) ↔ ⟨𝑋, (𝐹‘𝑋)⟩ ∈ 𝐹))
9		elrn2g 5725	. . . 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 297	. 2 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ (𝐹 “ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))
12	2, 11	syl5bir 246	1 ⊢ (𝑋 ∈ 𝑉 → ((𝐹‘𝑋) ∈ ran (𝐹 ↾ {𝑋}) → (𝐹‘𝑋) ∈ ran 𝐹))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3441  {csn 4525  ⟨cop 4531  ran crn 5520   ↾ cres 5521   “ cima 5522  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fv 6332

This theorem is referenced by:  fvn0fvelrn  6902  funressndmfvrn  43636