Theorem respreima 5220

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 4857	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		elin 3103	. . . . . . . . 9 ⊢ (x ∈ (B ∩ dom 𝐹) ↔ (x ∈ B ∧ x ∈ dom 𝐹))
3		ancom 253	. . . . . . . . 9 ⊢ ((x ∈ B ∧ x ∈ dom 𝐹) ↔ (x ∈ dom 𝐹 ∧ x ∈ B))
4	2, 3	bitri 173	. . . . . . . 8 ⊢ (x ∈ (B ∩ dom 𝐹) ↔ (x ∈ dom 𝐹 ∧ x ∈ B))
5	4	anbi1i 434	. . . . . . 7 ⊢ ((x ∈ (B ∩ dom 𝐹) ∧ ((𝐹 ↾ B)‘x) ∈ A) ↔ ((x ∈ dom 𝐹 ∧ x ∈ B) ∧ ((𝐹 ↾ B)‘x) ∈ A))
6		fvres 5123	. . . . . . . . . 10 ⊢ (x ∈ B → ((𝐹 ↾ B)‘x) = (𝐹‘x))
7	6	eleq1d 2088	. . . . . . . . 9 ⊢ (x ∈ B → (((𝐹 ↾ B)‘x) ∈ A ↔ (𝐹‘x) ∈ A))
8	7	adantl 262	. . . . . . . 8 ⊢ ((x ∈ dom 𝐹 ∧ x ∈ B) → (((𝐹 ↾ B)‘x) ∈ A ↔ (𝐹‘x) ∈ A))
9	8	pm5.32i 430	. . . . . . 7 ⊢ (((x ∈ dom 𝐹 ∧ x ∈ B) ∧ ((𝐹 ↾ B)‘x) ∈ A) ↔ ((x ∈ dom 𝐹 ∧ x ∈ B) ∧ (𝐹‘x) ∈ A))
10	5, 9	bitri 173	. . . . . 6 ⊢ ((x ∈ (B ∩ dom 𝐹) ∧ ((𝐹 ↾ B)‘x) ∈ A) ↔ ((x ∈ dom 𝐹 ∧ x ∈ B) ∧ (𝐹‘x) ∈ A))
11	10	a1i 9	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((x ∈ (B ∩ dom 𝐹) ∧ ((𝐹 ↾ B)‘x) ∈ A) ↔ ((x ∈ dom 𝐹 ∧ x ∈ B) ∧ (𝐹‘x) ∈ A)))
12		an32 484	. . . . 5 ⊢ (((x ∈ dom 𝐹 ∧ x ∈ B) ∧ (𝐹‘x) ∈ A) ↔ ((x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ A) ∧ x ∈ B))
13	11, 12	syl6bb 185	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((x ∈ (B ∩ dom 𝐹) ∧ ((𝐹 ↾ B)‘x) ∈ A) ↔ ((x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ A) ∧ x ∈ B)))
14		fnfun 4922	. . . . . . . 8 ⊢ (𝐹 Fn dom 𝐹 → Fun 𝐹)
15		funres 4867	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ B))
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → Fun (𝐹 ↾ B))
17		dmres 4559	. . . . . . 7 ⊢ dom (𝐹 ↾ B) = (B ∩ dom 𝐹)
18	16, 17	jctir 296	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (Fun (𝐹 ↾ B) ∧ dom (𝐹 ↾ B) = (B ∩ dom 𝐹)))
19		df-fn 4832	. . . . . 6 ⊢ ((𝐹 ↾ B) Fn (B ∩ dom 𝐹) ↔ (Fun (𝐹 ↾ B) ∧ dom (𝐹 ↾ B) = (B ∩ dom 𝐹)))
20	18, 19	sylibr 137	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ B) Fn (B ∩ dom 𝐹))
21		elpreima 5211	. . . . 5 ⊢ ((𝐹 ↾ B) Fn (B ∩ dom 𝐹) → (x ∈ (◡(𝐹 ↾ B) “ A) ↔ (x ∈ (B ∩ dom 𝐹) ∧ ((𝐹 ↾ B)‘x) ∈ A)))
22	20, 21	syl 14	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (x ∈ (◡(𝐹 ↾ B) “ A) ↔ (x ∈ (B ∩ dom 𝐹) ∧ ((𝐹 ↾ B)‘x) ∈ A)))
23		elin 3103	. . . . 5 ⊢ (x ∈ ((◡𝐹 “ A) ∩ B) ↔ (x ∈ (◡𝐹 “ A) ∧ x ∈ B))
24		elpreima 5211	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (x ∈ (◡𝐹 “ A) ↔ (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ A)))
25	24	anbi1d 441	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((x ∈ (◡𝐹 “ A) ∧ x ∈ B) ↔ ((x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ A) ∧ x ∈ B)))
26	23, 25	syl5bb 181	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (x ∈ ((◡𝐹 “ A) ∩ B) ↔ ((x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ A) ∧ x ∈ B)))
27	13, 22, 26	3bitr4d 209	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (x ∈ (◡(𝐹 ↾ B) “ A) ↔ x ∈ ((◡𝐹 “ A) ∩ B)))
28	1, 27	sylbi 114	. 2 ⊢ (Fun 𝐹 → (x ∈ (◡(𝐹 ↾ B) “ A) ↔ x ∈ ((◡𝐹 “ A) ∩ B)))
29	28	eqrdv 2020	1 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ B) “ A) = ((◡𝐹 “ A) ∩ B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1228   ∈ wcel 1374   ∩ cin 2893  ^◡ccnv 4271  dom cdm 4272   ↾ cres 4274   “ cima 4275  Fun wfun 4823   Fn wfn 4824  ‘cfv 4829

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837

This theorem is referenced by: (None)