Theorem wfximgfd 42901

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 6716	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		wfximgfd.1	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
4	2, 3, 3	fnfvimad 7233	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2107   “ cima 5679  ⟶wf 6537  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549

This theorem is referenced by:  imo72b2lem0  42903