Theorem fnfvimad 7190

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 4192	. . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2		imass2 6069	. . 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 4191	. . . 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 4154	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∩ 𝐶))
10		fnfvima 7189	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝐵 ∈ (𝐴 ∩ 𝐶)) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
11	4, 6, 9, 10	syl3anc 1374	. 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
12	3, 11	sselid 3933	1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3902   ⊆ wss 3903   “ cima 5635   Fn wfn 6495  ‘cfv 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-fv 6508

This theorem is referenced by:  cycpm3cl2  33229  rhmimaidl  33524  ig1pmindeg  33694  exsslsb  33773  dimkerim  33804  hashscontpow  42486  aks6d1c3  42487  aks6d1c2  42494  wfximgfd  44513  limsupmnflem  46072  liminfval2  46120  limsup10exlem  46124  liminflelimsupuz  46137  fundcmpsurinjimaid  47765