Theorem funfvima2d 7238

Description: A function's value in a preimage belongs to the image. (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by AV, 23-Mar-2024.)

Hypothesis

Ref	Expression
funfvima2d.1	⊢ (𝜑 → 𝐹:𝐴⟶𝐵)

Assertion

Ref	Expression
funfvima2d	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))

Proof of Theorem funfvima2d

Step	Hyp	Ref	Expression
1		funfvima2d.1	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵)
2	1	ffund 6714	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		ssidd 3931	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
4	1	fdmd 6720	. . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrrd 3945	. . 3 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
6		funfvima2 7237	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)))
7	2, 5, 6	syl2anc 596	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)))
8	7	imp 412	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144   ⊆ wss 3895  dom cdm 5650   “ cima 5653  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546

This theorem is used by:  imo72b2lem1  45262  fundcmpsurbijinjpreimafv  48582  imaid  50363  imaf1co  50364