Theorem elpreima 5408

Description: Membership in the preimage of a set under a function. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
elpreima	⊢ (F Fn A → (B ∈ (◡F “ C) ↔ (B ∈ A ∧ (F ‘B) ∈ C)))

Proof of Theorem elpreima

Step	Hyp	Ref	Expression
1		cnvimass 5017	. . . . 5 ⊢ (◡F “ C) ⊆ dom F
2	1	sseli 3270	. . . 4 ⊢ (B ∈ (◡F “ C) → B ∈ dom F)
3		fndm 5183	. . . . 5 ⊢ (F Fn A → dom F = A)
4	3	eleq2d 2420	. . . 4 ⊢ (F Fn A → (B ∈ dom F ↔ B ∈ A))
5	2, 4	syl5ib 210	. . 3 ⊢ (F Fn A → (B ∈ (◡F “ C) → B ∈ A))
6		fnfun 5182	. . . . 5 ⊢ (F Fn A → Fun F)
7		fvimacnvi 5403	. . . . 5 ⊢ ((Fun F ∧ B ∈ (◡F “ C)) → (F ‘B) ∈ C)
8	6, 7	sylan 457	. . . 4 ⊢ ((F Fn A ∧ B ∈ (◡F “ C)) → (F ‘B) ∈ C)
9	8	ex 423	. . 3 ⊢ (F Fn A → (B ∈ (◡F “ C) → (F ‘B) ∈ C))
10	5, 9	jcad 519	. 2 ⊢ (F Fn A → (B ∈ (◡F “ C) → (B ∈ A ∧ (F ‘B) ∈ C)))
11		fvimacnv 5404	. . . . 5 ⊢ ((Fun F ∧ B ∈ dom F) → ((F ‘B) ∈ C ↔ B ∈ (◡F “ C)))
12	11	funfni 5184	. . . 4 ⊢ ((F Fn A ∧ B ∈ A) → ((F ‘B) ∈ C ↔ B ∈ (◡F “ C)))
13	12	biimpd 198	. . 3 ⊢ ((F Fn A ∧ B ∈ A) → ((F ‘B) ∈ C → B ∈ (◡F “ C)))
14	13	expimpd 586	. 2 ⊢ (F Fn A → ((B ∈ A ∧ (F ‘B) ∈ C) → B ∈ (◡F “ C)))
15	10, 14	impbid 183	1 ⊢ (F Fn A → (B ∈ (◡F “ C) ↔ (B ∈ A ∧ (F ‘B) ∈ C)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∈ wcel 1710   “ 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:  unpreima  5409  inpreima  5410  respreima  5411