Theorem respreima 7057

Description: The preimage of a restricted function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
respreima	⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵))

Proof of Theorem respreima

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 6562	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		elin 3915	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹))
3	2	biancomi 468	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵))
4	3	anbi1i 636	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴))
5		fvres 6896	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
6	5	eleq1d 2846	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹‘𝑥) ∈ 𝐴))
7	6	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹‘𝑥) ∈ 𝐴))
8	7	pm5.32i 585	. . . . . . 7 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ (𝐹‘𝑥) ∈ 𝐴))
9	4, 8	bitri 278	. . . . . 6 ⊢ ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ (𝐹‘𝑥) ∈ 𝐴))
10	9	a1i 11	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ (𝐹‘𝑥) ∈ 𝐴)))
11		an32 659	. . . . 5 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ (𝐹‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵))
12	10, 11	bitrdi 290	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
13		fnfun 6631	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → Fun 𝐹)
14	13	funresd 6575	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → Fun (𝐹 ↾ 𝐵))
15		dmres 6003	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
16		df-fn 6534	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) Fn (𝐵 ∩ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)))
17	14, 15, 16	sylanblrc 602	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ 𝐵) Fn (𝐵 ∩ dom 𝐹))
18		elpreima 7049	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) Fn (𝐵 ∩ dom 𝐹) → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴)))
19	17, 18	syl 18	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴)))
20		elin 3915	. . . . 5 ⊢ (𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵) ↔ (𝑥 ∈ (◡𝐹 “ 𝐴) ∧ 𝑥 ∈ 𝐵))
21		elpreima 7049	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴)))
22	21	anbi1d 643	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (◡𝐹 “ 𝐴) ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
23	20, 22	bitrid 286	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
24	12, 19, 23	3bitr4d 314	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵)))
25	1, 24	sylbi 220	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵)))
26	25	eqrdv 2759	1 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ∩ cin 3898  ^◡ccnv 5650  dom cdm 5651   ↾ cres 5653   “ cima 5654  Fun wfun 6525   Fn wfn 6526  ‘cfv 6531

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 6487  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is used by:  paste  23592  restmetu  24869  eulerpartlemt  34986  smfres  47744