Theorem fvelimab 5370

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 2867	. . 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 5339	. . . . 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 2734	. . 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 2635	. . . . . 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 5181	. . . . . . 7 ⊢ (F Fn A → Fun F)
14	13	adantr 451	. . . . . 6 ⊢ ((F Fn A ∧ B ⊆ A) → Fun F)
15		fndm 5182	. . . . . . . 8 ⊢ (F Fn A → dom F = A)
16	15	sseq2d 3299	. . . . . . 7 ⊢ (F Fn A → (B ⊆ dom F ↔ B ⊆ A))
17	16	biimpar 471	. . . . . 6 ⊢ ((F Fn A ∧ B ⊆ A) → B ⊆ dom F)
18		dfimafn 5366	. . . . . 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 2462	. . . 4 ⊢ ((F Fn A ∧ B ⊆ A) → (y ∈ (F “ B) ↔ ∃x ∈ B (F ‘x) = y))
21	12, 20	vtoclg 2914	. . 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 2615  Vcvv 2859   ⊆ wss 3257   “ cima 4722  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-cnv 4785  df-rn 4786  df-dm 4787  df-fun 4789  df-fn 4790  df-fv 4795

This theorem is referenced by:  f1elima  5474  ovelimab  5610  dfnnc3  5885