Theorem fvimage 36130

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 3451	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		imaeq2 6016	. . 3 ⊢ (𝑥 = 𝐴 → (𝑅 “ 𝑥) = (𝑅 “ 𝐴))
3		imageval 36129	. . 3 ⊢ Image𝑅 = (𝑥 ∈ V ↦ (𝑅 “ 𝑥))
4	2, 3	fvmptg 6940	. 2 ⊢ ((𝐴 ∈ V ∧ (𝑅 “ 𝐴) ∈ 𝑊) → (Image𝑅‘𝐴) = (𝑅 “ 𝐴))
5	1, 4	sylan 581	1 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑅 “ 𝐴) ∈ 𝑊) → (Image𝑅‘𝐴) = (𝑅 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   “ cima 5628  ‘cfv 6493  Imagecimage 36039

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4194  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7936  df-2nd 7937  df-txp 36053  df-image 36063

This theorem is referenced by: (None)