Theorem respreima 5238

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 4874	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		elin 3120	. . . . . . . . 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 431	. . . . . . 7 ⊢ ((x ∈ (B ∩ dom 𝐹) ∧ ((𝐹 ↾ B)‘x) ∈ A) ↔ ((x ∈ dom 𝐹 ∧ x ∈ B) ∧ ((𝐹 ↾ B)‘x) ∈ A))
6		fvres 5141	. . . . . . . . . 10 ⊢ (x ∈ B → ((𝐹 ↾ B)‘x) = (𝐹‘x))
7	6	eleq1d 2103	. . . . . . . . 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 427	. . . . . . 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 496	. . . . 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 4939	. . . . . . . 8 ⊢ (𝐹 Fn dom 𝐹 → Fun 𝐹)
15		funres 4884	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ B))
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → Fun (𝐹 ↾ B))
17		dmres 4575	. . . . . . 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 4848	. . . . . 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 5229	. . . . 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 3120	. . . . 5 ⊢ (x ∈ ((◡𝐹 “ A) ∩ B) ↔ (x ∈ (◡𝐹 “ A) ∧ x ∈ B))
24		elpreima 5229	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (x ∈ (◡𝐹 “ A) ↔ (x ∈ dom 𝐹 ∧ (𝐹‘x) ∈ A)))
25	24	anbi1d 438	. . . . 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 2035	1 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ B) “ A) = ((◡𝐹 “ A) ∩ B))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390   ∩ cin 2910  ^◡ccnv 4287  dom cdm 4288   ↾ cres 4290   “ cima 4291  Fun wfun 4839   Fn wfn 4840  ‘cfv 4845

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by: (None)