Theorem respreima 5410

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 5136	. . 3 ⊢ (Fun F ↔ F Fn dom F)
2		elin 3219	. . . . . . . . 9 ⊢ (x ∈ (B ∩ dom F) ↔ (x ∈ B ∧ x ∈ dom F))
3		ancom 437	. . . . . . . . 9 ⊢ ((x ∈ B ∧ x ∈ dom F) ↔ (x ∈ dom F ∧ x ∈ B))
4	2, 3	bitri 240	. . . . . . . 8 ⊢ (x ∈ (B ∩ dom F) ↔ (x ∈ dom F ∧ x ∈ B))
5	4	anbi1i 676	. . . . . . 7 ⊢ ((x ∈ (B ∩ dom F) ∧ ((F ↾ B) ‘x) ∈ A) ↔ ((x ∈ dom F ∧ x ∈ B) ∧ ((F ↾ B) ‘x) ∈ A))
6		fvres 5342	. . . . . . . . . 10 ⊢ (x ∈ B → ((F ↾ B) ‘x) = (F ‘x))
7	6	eleq1d 2419	. . . . . . . . 9 ⊢ (x ∈ B → (((F ↾ B) ‘x) ∈ A ↔ (F ‘x) ∈ A))
8	7	adantl 452	. . . . . . . 8 ⊢ ((x ∈ dom F ∧ x ∈ B) → (((F ↾ B) ‘x) ∈ A ↔ (F ‘x) ∈ A))
9	8	pm5.32i 618	. . . . . . 7 ⊢ (((x ∈ dom F ∧ x ∈ B) ∧ ((F ↾ B) ‘x) ∈ A) ↔ ((x ∈ dom F ∧ x ∈ B) ∧ (F ‘x) ∈ A))
10	5, 9	bitri 240	. . . . . 6 ⊢ ((x ∈ (B ∩ dom F) ∧ ((F ↾ B) ‘x) ∈ A) ↔ ((x ∈ dom F ∧ x ∈ B) ∧ (F ‘x) ∈ A))
11	10	a1i 10	. . . . 5 ⊢ (F Fn dom F → ((x ∈ (B ∩ dom F) ∧ ((F ↾ B) ‘x) ∈ A) ↔ ((x ∈ dom F ∧ x ∈ B) ∧ (F ‘x) ∈ A)))
12		an32 773	. . . . 5 ⊢ (((x ∈ dom F ∧ x ∈ B) ∧ (F ‘x) ∈ A) ↔ ((x ∈ dom F ∧ (F ‘x) ∈ A) ∧ x ∈ B))
13	11, 12	syl6bb 252	. . . 4 ⊢ (F Fn dom F → ((x ∈ (B ∩ dom F) ∧ ((F ↾ B) ‘x) ∈ A) ↔ ((x ∈ dom F ∧ (F ‘x) ∈ A) ∧ x ∈ B)))
14		fnfun 5181	. . . . . . . 8 ⊢ (F Fn dom F → Fun F)
15		funres 5143	. . . . . . . 8 ⊢ (Fun F → Fun (F ↾ B))
16	14, 15	syl 15	. . . . . . 7 ⊢ (F Fn dom F → Fun (F ↾ B))
17		dmres 4986	. . . . . . 7 ⊢ dom (F ↾ B) = (B ∩ dom F)
18	16, 17	jctir 524	. . . . . 6 ⊢ (F Fn dom F → (Fun (F ↾ B) ∧ dom (F ↾ B) = (B ∩ dom F)))
19		df-fn 4790	. . . . . 6 ⊢ ((F ↾ B) Fn (B ∩ dom F) ↔ (Fun (F ↾ B) ∧ dom (F ↾ B) = (B ∩ dom F)))
20	18, 19	sylibr 203	. . . . 5 ⊢ (F Fn dom F → (F ↾ B) Fn (B ∩ dom F))
21		elpreima 5407	. . . . 5 ⊢ ((F ↾ B) Fn (B ∩ dom F) → (x ∈ (◡(F ↾ B) “ A) ↔ (x ∈ (B ∩ dom F) ∧ ((F ↾ B) ‘x) ∈ A)))
22	20, 21	syl 15	. . . 4 ⊢ (F Fn dom F → (x ∈ (◡(F ↾ B) “ A) ↔ (x ∈ (B ∩ dom F) ∧ ((F ↾ B) ‘x) ∈ A)))
23		elin 3219	. . . . 5 ⊢ (x ∈ ((◡F “ A) ∩ B) ↔ (x ∈ (◡F “ A) ∧ x ∈ B))
24		elpreima 5407	. . . . . 6 ⊢ (F Fn dom F → (x ∈ (◡F “ A) ↔ (x ∈ dom F ∧ (F ‘x) ∈ A)))
25	24	anbi1d 685	. . . . 5 ⊢ (F Fn dom F → ((x ∈ (◡F “ A) ∧ x ∈ B) ↔ ((x ∈ dom F ∧ (F ‘x) ∈ A) ∧ x ∈ B)))
26	23, 25	syl5bb 248	. . . 4 ⊢ (F Fn dom F → (x ∈ ((◡F “ A) ∩ B) ↔ ((x ∈ dom F ∧ (F ‘x) ∈ A) ∧ x ∈ B)))
27	13, 22, 26	3bitr4d 276	. . 3 ⊢ (F Fn dom F → (x ∈ (◡(F ↾ B) “ A) ↔ x ∈ ((◡F “ A) ∩ B)))
28	1, 27	sylbi 187	. 2 ⊢ (Fun F → (x ∈ (◡(F ↾ B) “ A) ↔ x ∈ ((◡F “ A) ∩ B)))
29	28	eqrdv 2351	1 ⊢ (Fun F → (◡(F ↾ B) “ A) = ((◡F “ A) ∩ B))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3208   “ cima 4722  ^◡ccnv 4771  dom cdm 4772   ↾ cres 4774  Fun wfun 4775   Fn wfn 4776   ‘cfv 4781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-co 4726  df-ima 4727  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-fv 4795

This theorem is referenced by: (None)