Theorem wfximgfd 44125

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 6748	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		wfximgfd.1	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
4	2, 3, 3	fnfvimad 7271	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2108   “ cima 5703  ⟶wf 6569  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581

This theorem is referenced by:  imo72b2lem0  44127