Theorem fvelimab 5371

Description: Function value in an image. (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (An unnecessary distinct variable restriction was removed by David Abernethy, 17-Dec-2011.) (Contributed by set.mm contributors, 20-Jan-2007.) (Revised by set.mm contributors, 25-Dec-2011.)

Assertion

Ref	Expression
fvelimab	⊢ ((F Fn A ∧ B ⊆ A) → (C ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = C))

Distinct variable groups:   x,B   x,C   x,F

Allowed substitution hint:   A(x)

Proof of Theorem fvelimab

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2868	. . 3 ⊢ (C ∈ (F “ B) → C ∈ V)
2	1	anim2i 552	. 2 ⊢ (((F Fn A ∧ B ⊆ A) ∧ C ∈ (F “ B)) → ((F Fn A ∧ B ⊆ A) ∧ C ∈ V))
3		fvex 5340	. . . . 5 ⊢ (F ‘x) ∈ V
4		eleq1 2413	. . . . 5 ⊢ ((F ‘x) = C → ((F ‘x) ∈ V ↔ C ∈ V))
5	3, 4	mpbii 202	. . . 4 ⊢ ((F ‘x) = C → C ∈ V)
6	5	rexlimivw 2735	. . 3 ⊢ (∃x ∈ B (F ‘x) = C → C ∈ V)
7	6	anim2i 552	. 2 ⊢ (((F Fn A ∧ B ⊆ A) ∧ ∃x ∈ B (F ‘x) = C) → ((F Fn A ∧ B ⊆ A) ∧ C ∈ V))
8		eleq1 2413	. . . . . 6 ⊢ (y = C → (y ∈ (F “ B) ↔ C ∈ (F “ B)))
9		eqeq2 2362	. . . . . . 7 ⊢ (y = C → ((F ‘x) = y ↔ (F ‘x) = C))
10	9	rexbidv 2636	. . . . . 6 ⊢ (y = C → (∃x ∈ B (F ‘x) = y ↔ ∃x ∈ B (F ‘x) = C))
11	8, 10	bibi12d 312	. . . . 5 ⊢ (y = C → ((y ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = y) ↔ (C ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = C)))
12	11	imbi2d 307	. . . 4 ⊢ (y = C → (((F Fn A ∧ B ⊆ A) → (y ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = y)) ↔ ((F Fn A ∧ B ⊆ A) → (C ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = C))))
13		fnfun 5182	. . . . . . 7 ⊢ (F Fn A → Fun F)
14	13	adantr 451	. . . . . 6 ⊢ ((F Fn A ∧ B ⊆ A) → Fun F)
15		fndm 5183	. . . . . . . 8 ⊢ (F Fn A → dom F = A)
16	15	sseq2d 3300	. . . . . . 7 ⊢ (F Fn A → (B ⊆ dom F ↔ B ⊆ A))
17	16	biimpar 471	. . . . . 6 ⊢ ((F Fn A ∧ B ⊆ A) → B ⊆ dom F)
18		dfimafn 5367	. . . . . 6 ⊢ ((Fun F ∧ B ⊆ dom F) → (F “ B) = {y ∣ ∃x ∈ B (F ‘x) = y})
19	14, 17, 18	syl2anc 642	. . . . 5 ⊢ ((F Fn A ∧ B ⊆ A) → (F “ B) = {y ∣ ∃x ∈ B (F ‘x) = y})
20	19	abeq2d 2463	. . . 4 ⊢ ((F Fn A ∧ B ⊆ A) → (y ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = y))
21	12, 20	vtoclg 2915	. . 3 ⊢ (C ∈ V → ((F Fn A ∧ B ⊆ A) → (C ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = C)))
22	21	impcom 419	. 2 ⊢ (((F Fn A ∧ B ⊆ A) ∧ C ∈ V) → (C ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = C))
23	2, 7, 22	pm5.21nd 868	1 ⊢ ((F Fn A ∧ B ⊆ A) → (C ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  ∃wrex 2616  Vcvv 2860   ⊆ wss 3258   “ cima 4723  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-cnv 4786  df-rn 4787  df-dm 4788  df-fun 4790  df-fn 4791  df-fv 4796

This theorem is referenced by:  f1elima  5475  ovelimab  5611  dfnnc3  5886