Theorem wfximgfd 44187

Description: The value of a function on its domain is in the image of the function. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypotheses

Ref	Expression
wfximgfd.1	⊢ (𝜑 → 𝐶 ∈ 𝐴)
wfximgfd.2	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
wfximgfd	⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴))

Proof of Theorem wfximgfd

Step	Hyp	Ref	Expression
1		wfximgfd.2	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffnd 6707	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		wfximgfd.1	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
4	2, 3, 3	fnfvimad 7226	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2108   “ cima 5657  ⟶wf 6527  ‘cfv 6531

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539

This theorem is referenced by:  imo72b2lem0  44189