Theorem resfvresima 7179

Description: The value of the function value of a restriction for a function restricted to the image of the restricting subset. (Contributed by AV, 6-Mar-2021.)

Hypotheses

Ref	Expression
resfvresima.f	⊢ (𝜑 → Fun 𝐹)
resfvresima.s	⊢ (𝜑 → 𝑆 ⊆ dom 𝐹)
resfvresima.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Assertion

Ref	Expression
resfvresima	⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋)))

Proof of Theorem resfvresima

Step	Hyp	Ref	Expression
1		resfvresima.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
2	1	fvresd 6852	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘𝑋) = (𝐹‘𝑋))
3	2	fveq2d 6836	. 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋)))
4		resfvresima.f	. . . . 5 ⊢ (𝜑 → Fun 𝐹)
5		resfvresima.s	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ dom 𝐹)
6	4, 5	jca 511	. . . 4 ⊢ (𝜑 → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹))
7		funfvima2 7175	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)))
8	6, 1, 7	sylc 65	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))
9	8	fvresd 6852	. 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋)) = (𝐻‘(𝐹‘𝑋)))
10	3, 9	eqtrd 2769	1 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ⊆ wss 3899  dom cdm 5622   ↾ cres 5624   “ cima 5625  Fun wfun 6484  ‘cfv 6490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498

This theorem is referenced by:  wlkres  29691