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Theorem fvelimab 6730
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 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 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.
StepHypRef Expression
1 elex 3510 . . 3 (𝐶 ∈ (𝐹𝐵) → 𝐶 ∈ V)
21anim2i 616 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ (𝐹𝐵)) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
3 fvex 6676 . . . . 5 (𝐹𝑥) ∈ V
4 eleq1 2897 . . . . 5 ((𝐹𝑥) = 𝐶 → ((𝐹𝑥) ∈ V ↔ 𝐶 ∈ V))
53, 4mpbii 234 . . . 4 ((𝐹𝑥) = 𝐶𝐶 ∈ V)
65rexlimivw 3279 . . 3 (∃𝑥𝐵 (𝐹𝑥) = 𝐶𝐶 ∈ V)
76anim2i 616 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ ∃𝑥𝐵 (𝐹𝑥) = 𝐶) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
8 eleq1 2897 . . . . . 6 (𝑦 = 𝐶 → (𝑦 ∈ (𝐹𝐵) ↔ 𝐶 ∈ (𝐹𝐵)))
9 eqeq2 2830 . . . . . . 7 (𝑦 = 𝐶 → ((𝐹𝑥) = 𝑦 ↔ (𝐹𝑥) = 𝐶))
109rexbidv 3294 . . . . . 6 (𝑦 = 𝐶 → (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
118, 10bibi12d 347 . . . . 5 (𝑦 = 𝐶 → ((𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦) ↔ (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)))
1211imbi2d 342 . . . 4 (𝑦 = 𝐶 → (((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))))
13 fnfun 6446 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
14 fndm 6448 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
1514sseq2d 3996 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
1615biimpar 478 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
17 dfimafn 6721 . . . . . 6 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1813, 16, 17syl2an2r 681 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1918abeq2d 2944 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
2012, 19vtoclg 3565 . . 3 (𝐶 ∈ V → ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)))
2120impcom 408 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
222, 7, 21pm5.21nd 798 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {cab 2796  wrex 3136  Vcvv 3492  wss 3933  dom cdm 5548  cima 5551  Fun wfun 6342   Fn wfn 6343  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-fv 6356
This theorem is referenced by:  fvelimabd  6731  unima  6732  ssimaex  6741  rexima  6990  ralima  6991  f1elima  7012  ovelimab  7315  fimaproj  7818  tcrank  9301  djuun  9343  ackbij2  9653  fin1a2lem6  9815  iunfo  9949  grothomex  10239  axpre-sup  10579  injresinjlem  13145  txkgen  22188  fmucndlem  22827  efopn  25168  pjimai  29880  fimarab  30318  qtophaus  30999  indf1ofs  31184  eulerpartgbij  31529  eulerpartlemgvv  31533  ballotlemsima  31672  elmthm  32720  elintfv  32904  nocvxmin  33145  isnacs2  39181  isnacs3  39185  islmodfg  39547  kercvrlsm  39561  isnumbasgrplem2  39582  dfacbasgrp  39586  fourierdlem62  42330
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