Theorem resfvresima 7241

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 6905	. . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆)‘𝑋) = (𝐹‘𝑋))
3	2	fveq2d 6889	. 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 521	. . . 4 ⊢ (𝜑 → (Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹))
7		funfvima2 7237	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝑆 ⊆ dom 𝐹) → (𝑋 ∈ 𝑆 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆)))
8	6, 1, 7	sylc 66	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐹 “ 𝑆))
9	8	fvresd 6905	. 2 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘(𝐹‘𝑋)) = (𝐻‘(𝐹‘𝑋)))
10	3, 9	eqtrd 2796	1 ⊢ (𝜑 → ((𝐻 ↾ (𝐹 “ 𝑆))‘((𝐹 ↾ 𝑆)‘𝑋)) = (𝐻‘(𝐹‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  dom cdm 5651   ↾ cres 5653   “ cima 5654  Fun wfun 6532  ‘cfv 6538

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 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  wlkres  30249