Theorem funfvima2d 7239

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 6720	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		ssidd 3996	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝐴)
4	1	fdmd 6727	. . . 4 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	3, 4	sseqtrrd 4014	. . 3 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐹)
6		funfvima2 7238	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)))
7	2, 5, 6	syl2anc 582	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐴 → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴)))
8	7	imp 405	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐴) → (𝐹‘𝑋) ∈ (𝐹 “ 𝐴))

Syntax hints:   → wi 4   ∧ wa 394   ∈ wcel 2098   ⊆ wss 3940  dom cdm 5672   “ cima 5675  Fun wfun 6536  ⟶wf 6538  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550

This theorem is referenced by:  imo72b2lem1  43663  fundcmpsurbijinjpreimafv  46809