Theorem funfvima2d 6988

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 6512	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		ssidd 3989	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
4	1	fdmd 6517	. . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrrd 4007	. . 3 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
6		funfvima2 6987	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)))
7	2, 5, 6	syl2anc 586	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)))
8	7	imp 409	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2110   ⊆ wss 3935  dom cdm 5549   “ cima 5552  Fun wfun 6343  ⟶wf 6345  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357

This theorem is referenced by:  imo72b2lem1  40514  fundcmpsurbijinjpreimafv  43561