Theorem wfximgfd 44608

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 6663	. 2 ⊢ (𝜑 → 𝐹 Fn 𝐴)
3		wfximgfd.1	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
4	2, 3, 3	fnfvimad 7185	1 ⊢ (𝜑 → (𝐹‘𝐶) ∈ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∈ wcel 2119   “ cima 5628  ⟶wf 6488  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  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 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500

This theorem is referenced by:  imo72b2lem0  44610