Theorem respreima 7011

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 6522	. . 3 ⊢ (Fun 𝐹 ↔ 𝐹 Fn dom 𝐹)
2		elin 3917	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ 𝐵 ∧ 𝑥 ∈ dom 𝐹))
3	2	biancomi 462	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ↔ (𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵))
4	3	anbi1i 624	. . . . . . 7 ⊢ ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴))
5		fvres 6853	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → ((𝐹 ↾ 𝐵)‘𝑥) = (𝐹‘𝑥))
6	5	eleq1d 2821	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹‘𝑥) ∈ 𝐴))
7	6	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) → (((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴 ↔ (𝐹‘𝑥) ∈ 𝐴))
8	7	pm5.32i 574	. . . . . . 7 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ (𝐹‘𝑥) ∈ 𝐴))
9	4, 8	bitri 275	. . . . . 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 646	. . . . 5 ⊢ (((𝑥 ∈ dom 𝐹 ∧ 𝑥 ∈ 𝐵) ∧ (𝐹‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵))
12	10, 11	bitrdi 287	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
13		fnfun 6592	. . . . . . 7 ⊢ (𝐹 Fn dom 𝐹 → Fun 𝐹)
14	13	funresd 6535	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → Fun (𝐹 ↾ 𝐵))
15		dmres 5971	. . . . . 6 ⊢ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)
16		df-fn 6495	. . . . . 6 ⊢ ((𝐹 ↾ 𝐵) Fn (𝐵 ∩ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝐵) ∧ dom (𝐹 ↾ 𝐵) = (𝐵 ∩ dom 𝐹)))
17	14, 15, 16	sylanblrc 590	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ 𝐵) Fn (𝐵 ∩ dom 𝐹))
18		elpreima 7003	. . . . 5 ⊢ ((𝐹 ↾ 𝐵) Fn (𝐵 ∩ dom 𝐹) → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴)))
19	17, 18	syl 17	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ (𝑥 ∈ (𝐵 ∩ dom 𝐹) ∧ ((𝐹 ↾ 𝐵)‘𝑥) ∈ 𝐴)))
20		elin 3917	. . . . 5 ⊢ (𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵) ↔ (𝑥 ∈ (◡𝐹 “ 𝐴) ∧ 𝑥 ∈ 𝐵))
21		elpreima 7003	. . . . . 6 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡𝐹 “ 𝐴) ↔ (𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴)))
22	21	anbi1d 631	. . . . 5 ⊢ (𝐹 Fn dom 𝐹 → ((𝑥 ∈ (◡𝐹 “ 𝐴) ∧ 𝑥 ∈ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
23	20, 22	bitrid 283	. . . 4 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵) ↔ ((𝑥 ∈ dom 𝐹 ∧ (𝐹‘𝑥) ∈ 𝐴) ∧ 𝑥 ∈ 𝐵)))
24	12, 19, 23	3bitr4d 311	. . 3 ⊢ (𝐹 Fn dom 𝐹 → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵)))
25	1, 24	sylbi 217	. 2 ⊢ (Fun 𝐹 → (𝑥 ∈ (◡(𝐹 ↾ 𝐵) “ 𝐴) ↔ 𝑥 ∈ ((◡𝐹 “ 𝐴) ∩ 𝐵)))
26	25	eqrdv 2734	1 ⊢ (Fun 𝐹 → (◡(𝐹 ↾ 𝐵) “ 𝐴) = ((◡𝐹 “ 𝐴) ∩ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∩ cin 3900  ^◡ccnv 5623  dom cdm 5624   ↾ cres 5626   “ cima 5627  Fun wfun 6486   Fn wfn 6487  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  paste  23238  restmetu  24514  eulerpartlemt  34528  smfres  47030