Theorem resfvresima 7111

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 6794	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘𝑋) = (𝐹‘𝑋))
3	2	fveq2d 6778	. 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 512	. . . 4 ⊢ (𝜑 → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹))
7		funfvima2 7107	. . . 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 6794	. 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋)) = (𝐻‘(𝐹‘𝑋)))
10	3, 9	eqtrd 2778	1 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  dom cdm 5589   ↾ cres 5591   “ cima 5592  Fun wfun 6427  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  wlkres  28038