Theorem fvelimab 6914

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 3463	. . 3 ⊢ (𝐶 ∈ (𝐹 “ 𝐵) → 𝐶 ∈ V)
2	1	anim2i 617	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ (𝐹 “ 𝐵)) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
3		fvex 6855	. . . . 5 ⊢ (𝐹‘𝑥) ∈ V
4		eleq1 2825	. . . . 5 ⊢ ((𝐹‘𝑥) = 𝐶 → ((𝐹‘𝑥) ∈ V ↔ 𝐶 ∈ V))
5	3, 4	mpbii 232	. . . 4 ⊢ ((𝐹‘𝑥) = 𝐶 → 𝐶 ∈ V)
6	5	rexlimivw 3148	. . 3 ⊢ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶 → 𝐶 ∈ V)
7	6	anim2i 617	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶) → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V))
8		eleq1 2825	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ∈ (𝐹 “ 𝐵) ↔ 𝐶 ∈ (𝐹 “ 𝐵)))
9		eqeq2 2748	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ((𝐹‘𝑥) = 𝑦 ↔ (𝐹‘𝑥) = 𝐶))
10	9	rexbidv 3175	. . . . . 6 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))
11	8, 10	bibi12d 345	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦) ↔ (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)))
12	11	imbi2d 340	. . . 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 6602	. . . . . 6 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹)
14		fndm 6605	. . . . . . . 8 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
15	14	sseq2d 3976	. . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹 ↔ 𝐵 ⊆ 𝐴))
16	15	biimpar 478	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ dom 𝐹)
17		dfimafn 6905	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐵 ⊆ dom 𝐹) → (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦})
18	13, 16, 17	syl2an2r 683	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐹 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦})
19	18	eqabd 2880	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
20	12, 19	vtoclg 3525	. . 3 ⊢ (𝐶 ∈ V → ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶)))
21	20	impcom 408	. 2 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))
22	2, 7, 21	pm5.21nd 800	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐶))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2713  ∃wrex 3073  Vcvv 3445   ⊆ wss 3910  dom cdm 5633   “ cima 5636  Fun wfun 6490   Fn wfn 6491  ‘cfv 6496

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-fv 6504

This theorem is referenced by:  fvelimabd  6915  unima  6916  ssimaex  6926  rexima  7187  ralima  7188  f1elima  7210  fnssintima  7307  imaeqsexv  7308  ovelimab  7532  fimaproj  8067  tcrank  9820  djuun  9862  ackbij2  10179  fin1a2lem6  10341  iunfo  10475  grothomex  10765  axpre-sup  11105  injresinjlem  13692  txkgen  23003  fmucndlem  23643  efopn  26013  nocvxmin  27118  eqscut2  27145  cuteq0  27171  elold  27199  lrrecfr  27255  negsproplem2  27327  negsunif  27350  pjimai  31118  fimarab  31559  qtophaus  32417  indf1ofs  32625  eulerpartgbij  32972  eulerpartlemgvv  32976  ballotlemsima  33115  elmthm  34170  elintfv  34339  isnacs2  41015  isnacs3  41019  islmodfg  41382  kercvrlsm  41396  isnumbasgrplem2  41417  dfacbasgrp  41421  fourierdlem62  44399