Theorem respreima 5646

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 5248	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		elin 3320	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹))
3		ancom 266	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵))
4	2, 3	bitri 184	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵))
5	4	anbi1i 458	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴))
6		fvres 5541	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
7	6	eleq1d 2246	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹‘𝑥) ∈ 𝐴))
8	7	adantl 277	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹‘𝑥) ∈ 𝐴))
9	8	pm5.32i 454	. . . . . . 7 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ (𝐹‘𝑥) ∈ 𝐴))
10	5, 9	bitri 184	. . . . . 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 562	. . . . 5 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ (𝐹‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵))
13	11, 12	bitrdi 196	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
14		fnfun 5315	. . . . . . . 8 ⊢ (𝐹 Fn dom 𝐹 → Fun 𝐹)
15		funres 5259	. . . . . . . 8 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝐵))
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → Fun (𝐹 ↾ 𝐵))
17		dmres 4930	. . . . . . 7 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
18	16, 17	jctir 313	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)))
19		df-fn 5221	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) Fn (𝐵 ∩ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)))
20	18, 19	sylibr 134	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ 𝐵) Fn (𝐵 ∩ dom 𝐹))
21		elpreima 5637	. . . . 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 3320	. . . . 5 ⊢ (𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵) ↔ (𝑥 ∈ (◡𝐹 “ 𝐴) ∧ 𝑥 ∈ 𝐵))
24		elpreima 5637	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴)))
25	24	anbi1d 465	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (◡𝐹 “ 𝐴) ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
26	23, 25	bitrid 192	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
27	13, 22, 26	3bitr4d 220	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵)))
28	1, 27	sylbi 121	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵)))
29	28	eqrdv 2175	1 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148   ∩ cin 3130  ^◡ccnv 4627  dom cdm 4628   ↾ cres 4630   “ cima 4631  Fun wfun 5212   Fn wfn 5213  ‘cfv 5218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by: (None)