Theorem elpreima 5407

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 5016	. . . . 5 ⊢ (◡F “ C) ⊆ dom F
2	1	sseli 3269	. . . 4 ⊢ (B ∈ (◡F “ C) → B ∈ dom F)
3		fndm 5182	. . . . 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 5181	. . . . 5 ⊢ (F Fn A → Fun F)
7		fvimacnvi 5402	. . . . 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 5403	. . . . 5 ⊢ ((Fun F ∧ B ∈ dom F) → ((F ‘B) ∈ C ↔ B ∈ (◡F “ C)))
12	11	funfni 5183	. . . 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 4722  ^◡ccnv 4771  dom cdm 4772  Fun wfun 4775   Fn wfn 4776   ‘cfv 4781

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  unpreima  5408  inpreima  5409  respreima  5410