Theorem inpreima 5410

Description: Preimage of an intersection. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
inpreima	⊢ (Fun F → (◡F “ (A ∩ B)) = ((◡F “ A) ∩ (◡F “ B)))

Proof of Theorem inpreima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 5137	. 2 ⊢ (Fun F ↔ F Fn dom F)
2		elin 3220	. . . . . . . 8 ⊢ ((F ‘x) ∈ (A ∩ B) ↔ ((F ‘x) ∈ A ∧ (F ‘x) ∈ B))
3	2	anbi2i 675	. . . . . . 7 ⊢ ((x ∈ dom F ∧ (F ‘x) ∈ (A ∩ B)) ↔ (x ∈ dom F ∧ ((F ‘x) ∈ A ∧ (F ‘x) ∈ B)))
4	3	a1i 10	. . . . . 6 ⊢ (F Fn dom F → ((x ∈ dom F ∧ (F ‘x) ∈ (A ∩ B)) ↔ (x ∈ dom F ∧ ((F ‘x) ∈ A ∧ (F ‘x) ∈ B))))
5		anandi 801	. . . . . 6 ⊢ ((x ∈ dom F ∧ ((F ‘x) ∈ A ∧ (F ‘x) ∈ B)) ↔ ((x ∈ dom F ∧ (F ‘x) ∈ A) ∧ (x ∈ dom F ∧ (F ‘x) ∈ B)))
6	4, 5	syl6bb 252	. . . . 5 ⊢ (F Fn dom F → ((x ∈ dom F ∧ (F ‘x) ∈ (A ∩ B)) ↔ ((x ∈ dom F ∧ (F ‘x) ∈ A) ∧ (x ∈ dom F ∧ (F ‘x) ∈ B))))
7		elpreima 5408	. . . . 5 ⊢ (F Fn dom F → (x ∈ (◡F “ (A ∩ B)) ↔ (x ∈ dom F ∧ (F ‘x) ∈ (A ∩ B))))
8		elpreima 5408	. . . . . 6 ⊢ (F Fn dom F → (x ∈ (◡F “ A) ↔ (x ∈ dom F ∧ (F ‘x) ∈ A)))
9		elpreima 5408	. . . . . 6 ⊢ (F Fn dom F → (x ∈ (◡F “ B) ↔ (x ∈ dom F ∧ (F ‘x) ∈ B)))
10	8, 9	anbi12d 691	. . . . 5 ⊢ (F Fn dom F → ((x ∈ (◡F “ A) ∧ x ∈ (◡F “ B)) ↔ ((x ∈ dom F ∧ (F ‘x) ∈ A) ∧ (x ∈ dom F ∧ (F ‘x) ∈ B))))
11	6, 7, 10	3bitr4d 276	. . . 4 ⊢ (F Fn dom F → (x ∈ (◡F “ (A ∩ B)) ↔ (x ∈ (◡F “ A) ∧ x ∈ (◡F “ B))))
12		elin 3220	. . . 4 ⊢ (x ∈ ((◡F “ A) ∩ (◡F “ B)) ↔ (x ∈ (◡F “ A) ∧ x ∈ (◡F “ B)))
13	11, 12	syl6bbr 254	. . 3 ⊢ (F Fn dom F → (x ∈ (◡F “ (A ∩ B)) ↔ x ∈ ((◡F “ A) ∩ (◡F “ B))))
14	13	eqrdv 2351	. 2 ⊢ (F Fn dom F → (◡F “ (A ∩ B)) = ((◡F “ A) ∩ (◡F “ B)))
15	1, 14	sylbi 187	1 ⊢ (Fun F → (◡F “ (A ∩ B)) = ((◡F “ A) ∩ (◡F “ B)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∩ cin 3209   “ cima 4723  ^◡ccnv 4772  dom cdm 4773  Fun wfun 4776   Fn wfn 4777   ‘cfv 4782

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by: (None)