Theorem fnfvimad 5890

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 3428	. . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐶
2		imass2 5112	. . 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 3427	. . . 4 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴
6	5	a1i 9	. . 3 ⊢ (𝜑 → (𝐴 ∩ 𝐶) ⊆ 𝐴)
7		fnfvimad.2	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
8		fnfvimad.3	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
9	7, 8	elind 3392	. . 3 ⊢ (𝜑 → 𝐵 ∈ (𝐴 ∩ 𝐶))
10		fnfvima 5889	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴 ∧ 𝐵 ∈ (𝐴 ∩ 𝐶)) → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
11	4, 6, 9, 10	syl3anc 1273	. 2 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ (𝐴 ∩ 𝐶)))
12	3, 11	sselid 3225	1 ⊢ (𝜑 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐶))

Syntax hints:   → wi 4   ∈ wcel 2202   ∩ cin 3199   ⊆ wss 3200   “ cima 4728   Fn wfn 5321  ‘cfv 5326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334

This theorem is referenced by: (None)