Theorem fvimage 36421

Description: Value of the image functor. (Contributed by Scott Fenton, 4-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fvimage	⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅 “ 𝐴) ∈ 𝑊) → (Image𝑅‘𝐴) = (𝑅 “ 𝐴))

Proof of Theorem fvimage

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3476	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		imaeq2 6058	. . 3 ⊢ (𝑥 = 𝐴 → (𝑅 “ 𝑥) = (𝑅 “ 𝐴))
3		imageval 36420	. . 3 ⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
4	2, 3	fvmptg 6987	. 2 ⊢ ((𝐴 ∈ V ∧ (𝑅 “ 𝐴) ∈ 𝑊) → (Image𝑅‘𝐴) = (𝑅 “ 𝐴))
5	1, 4	sylan 591	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅 “ 𝐴) ∈ 𝑊) → (Image𝑅‘𝐴) = (𝑅 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  Vcvv 3455   “ cima 5664  ‘cfv 6536  Imagecimage 36330

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-symdif 4206  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-1st 7982  df-2nd 7983  df-txp 36344  df-image 36354

This theorem is referenced by: (None)