Theorem fnfvimad 7232

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 4183	. . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2		imass2 6096	. . 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 4182	. . . 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 4146	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∩ 𝐶))
10		fnfvima 7231	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝐵 ∈ (𝐴 ∩ 𝐶)) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
11	4, 6, 9, 10	syl3anc 1398	. 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
12	3, 11	sselid 3929	1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899   “ cima 5654   Fn wfn 6526  ‘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-fv 6539

This theorem is used by:  cycpm3cl2  33679  rhmimaidl  33964  ig1pmindeg  34116  exsslsb  34211  dimkerim  34241  hashscontpow  43140  aks6d1c3  43141  aks6d1c2  43148  wfximgfd  45122  limsupmnflem  46674  liminfval2  46722  limsup10exlem  46726  liminflelimsupuz  46739  fundcmpsurinjimaid  48437