Theorem fvelimab 6292

Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-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 3243	. . 3 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → 𝐶 ∈ V)
2	1	anim2i 592	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ (𝐹 “ 𝐵)) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
3		fvex 6239	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
4		eleq1 2718	. . . . 5 ⊢ ((𝐹‘𝑥) = 𝐶 → ((𝐹‘𝑥) ∈ V ↔ 𝐶 ∈ V))
5	3, 4	mpbii 223	. . . 4 ⊢ ((𝐹‘𝑥) = 𝐶 → 𝐶 ∈ V)
6	5	rexlimivw 3058	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶 → 𝐶 ∈ V)
7	6	anim2i 592	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
8		eleq1 2718	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ (𝐹 “ 𝐵) ↔ 𝐶 ∈ (𝐹 “ 𝐵)))
9		eqeq2 2662	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ((𝐹‘𝑥) = 𝑦 ↔ (𝐹‘𝑥) = 𝐶))
10	9	rexbidv 3081	. . . . . 6 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))
11	8, 10	bibi12d 334	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦) ↔ (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)))
12	11	imbi2d 329	. . . 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 6026	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
14		fndm 6028	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
15	14	sseq2d 3666	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
16	15	biimpar 501	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
17		dfimafn 6284	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦})
18	13, 16, 17	syl2an2r 893	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦})
19	18	abeq2d 2763	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
20	12, 19	vtoclg 3297	. . 3 ⊢ (𝐶 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)))
21	20	impcom 445	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))
22	2, 7, 21	pm5.21nd 961	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030  {cab 2637  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  dom cdm 5143   “ cima 5146  Fun wfun 5920   Fn wfn 5921  ‘cfv 5926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934

This theorem is referenced by:  fvelimabd  6293  ssimaex  6302  rexima  6537  ralima  6538  f1elima  6560  ovelimab  6854  tcrank  8785  ackbij2  9103  fin1a2lem6  9265  iunfo  9399  grothomex  9689  axpre-sup  10028  injresinjlem  12628  lmhmima  19095  txkgen  21503  fmucndlem  22142  mdegldg  23871  ig1peu  23976  efopn  24449  pjimai  29163  fimarab  29573  fimaproj  30028  qtophaus  30031  indf1ofs  30216  eulerpartgbij  30562  eulerpartlemgvv  30566  ballotlemsima  30705  elmthm  31599  nocvxmin  32019  isnacs2  37586  isnacs3  37590  islmodfg  37956  kercvrlsm  37970  isnumbasgrplem2  37991  dfacbasgrp  37995  unima  39660  fourierdlem62  40703