Theorem fnfvimad 39459

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 3834	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2		imass2 5501	. . . 4 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐶 → (𝐹 “ (𝐴 ∩ 𝐶)) ⊆ (𝐹 “ 𝐶))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝐹 “ (𝐴 ∩ 𝐶)) ⊆ (𝐹 “ 𝐶)
4	3	a1i 11	. 2 ⊢ (𝜑 → (𝐹 “ (𝐴 ∩ 𝐶)) ⊆ (𝐹 “ 𝐶))
5		fnfvimad.1	. . . . 5 ⊢ (𝜑 → 𝐹 Fn 𝐴)
6		fnfun 5988	. . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
7	5, 6	syl 17	. . . 4 ⊢ (𝜑 → Fun 𝐹)
8		inss1 3833	. . . . . 6 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴
9	8	a1i 11	. . . . 5 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴)
10	5	fndmd 39441	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
11	9, 10	sseqtr4d 3642	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ dom 𝐹)
12	7, 11	jca 554	. . 3 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐴 ∩ 𝐶) ⊆ dom 𝐹))
13		fnfvimad.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
14		fnfvimad.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
15	13, 14	elind 3798	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∩ 𝐶))
16		funfvima2 6493	. . 3 ⊢ ((Fun 𝐹 ∧ (𝐴 ∩ 𝐶) ⊆ dom 𝐹) → (𝐵 ∈ (𝐴 ∩ 𝐶) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶))))
17	12, 15, 16	sylc 65	. 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
18	4, 17	sseldd 3604	1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∧ wa 384   ∈ wcel 1990   ∩ cin 3573   ⊆ wss 3574  dom cdm 5114   “ cima 5117  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  limsupmnflem  39952  liminfval2  40000