Theorem fnfvimad 7271

Description: A function's value belongs to the image. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
fnfvimad.1	⊢ (𝜑 → 𝐹 Fn 𝐴)
fnfvimad.2	⊢ (𝜑 → 𝐵 ∈ 𝐴)
fnfvimad.3	⊢ (𝜑 → 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
fnfvimad	⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶))

Proof of Theorem fnfvimad

Step	Hyp	Ref	Expression
1		inss2 4259	. . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2		imass2 6132	. . 3 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 → (𝐹 “ (𝐴 ∩ 𝐶)) ⊆ (𝐹 “ 𝐶))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐹 “ (𝐴 ∩ 𝐶)) ⊆ (𝐹 “ 𝐶)
4		fnfvimad.1	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐴)
5		inss1 4258	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴
6	5	a1i 11	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴)
7		fnfvimad.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
8		fnfvimad.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
9	7, 8	elind 4223	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∩ 𝐶))
10		fnfvima 7270	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝐵 ∈ (𝐴 ∩ 𝐶)) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
11	4, 6, 9, 10	syl3anc 1371	. 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
12	3, 11	sselid 4006	1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∈ wcel 2108   ∩ cin 3975   ⊆ wss 3976   “ cima 5703   Fn wfn 6568  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581

This theorem is referenced by:  cycpm3cl2  33129  rhmimaidl  33425  ig1pmindeg  33587  dimkerim  33640  hashscontpow  42079  aks6d1c3  42080  aks6d1c2  42087  wfximgfd  44125  limsupmnflem  45641  liminfval2  45689  limsup10exlem  45693  liminflelimsupuz  45706  fundcmpsurinjimaid  47285