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Theorem fvelimab 6731
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 3513 . . 3 (𝐶 ∈ (𝐹𝐵) → 𝐶 ∈ V)
21anim2i 616 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ (𝐹𝐵)) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
3 fvex 6677 . . . . 5 (𝐹𝑥) ∈ V
4 eleq1 2900 . . . . 5 ((𝐹𝑥) = 𝐶 → ((𝐹𝑥) ∈ V ↔ 𝐶 ∈ V))
53, 4mpbii 234 . . . 4 ((𝐹𝑥) = 𝐶𝐶 ∈ V)
65rexlimivw 3282 . . 3 (∃𝑥𝐵 (𝐹𝑥) = 𝐶𝐶 ∈ V)
76anim2i 616 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ ∃𝑥𝐵 (𝐹𝑥) = 𝐶) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
8 eleq1 2900 . . . . . 6 (𝑦 = 𝐶 → (𝑦 ∈ (𝐹𝐵) ↔ 𝐶 ∈ (𝐹𝐵)))
9 eqeq2 2833 . . . . . . 7 (𝑦 = 𝐶 → ((𝐹𝑥) = 𝑦 ↔ (𝐹𝑥) = 𝐶))
109rexbidv 3297 . . . . . 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 6447 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
14 fndm 6449 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
1514sseq2d 3998 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
1615biimpar 478 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
17 dfimafn 6722 . . . . . 6 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1813, 16, 17syl2an2r 681 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1918abeq2d 2947 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
2012, 19vtoclg 3568 . . 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 2799  wrex 3139  Vcvv 3495  wss 3935  dom cdm 5549  cima 5552  Fun wfun 6343   Fn wfn 6344  cfv 6349
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 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357
This theorem is referenced by:  fvelimabd  6732  unima  6733  ssimaex  6742  rexima  6990  ralima  6991  f1elima  7012  ovelimab  7315  fimaproj  7820  tcrank  9302  djuun  9344  ackbij2  9654  fin1a2lem6  9816  iunfo  9950  grothomex  10240  axpre-sup  10580  injresinjlem  13147  txkgen  22190  fmucndlem  22829  efopn  25168  pjimai  29881  fimarab  30319  qtophaus  31000  indf1ofs  31185  eulerpartgbij  31530  eulerpartlemgvv  31534  ballotlemsima  31673  elmthm  32721  elintfv  32905  nocvxmin  33146  isnacs2  39183  isnacs3  39187  islmodfg  39549  kercvrlsm  39563  isnumbasgrplem2  39584  dfacbasgrp  39588  fourierdlem62  42334
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