Theorem funfvima2d 7230

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 6706	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		ssidd 3954	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
4	1	fdmd 6712	. . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrrd 3968	. . 3 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
6		funfvima2 7229	. . 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 2145   ⊆ wss 3899  dom cdm 5651   “ cima 5654  Fun wfun 6525  ⟶wf 6527  ‘cfv 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539

This theorem is used by:  imo72b2lem1  45128  fundcmpsurbijinjpreimafv  48433  imaid  50206  imaf1co  50207