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Theorem fvelimab 6712
 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 3459 . . 3 (𝐶 ∈ (𝐹𝐵) → 𝐶 ∈ V)
21anim2i 619 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ (𝐹𝐵)) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
3 fvex 6658 . . . . 5 (𝐹𝑥) ∈ V
4 eleq1 2877 . . . . 5 ((𝐹𝑥) = 𝐶 → ((𝐹𝑥) ∈ V ↔ 𝐶 ∈ V))
53, 4mpbii 236 . . . 4 ((𝐹𝑥) = 𝐶𝐶 ∈ V)
65rexlimivw 3241 . . 3 (∃𝑥𝐵 (𝐹𝑥) = 𝐶𝐶 ∈ V)
76anim2i 619 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ ∃𝑥𝐵 (𝐹𝑥) = 𝐶) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
8 eleq1 2877 . . . . . 6 (𝑦 = 𝐶 → (𝑦 ∈ (𝐹𝐵) ↔ 𝐶 ∈ (𝐹𝐵)))
9 eqeq2 2810 . . . . . . 7 (𝑦 = 𝐶 → ((𝐹𝑥) = 𝑦 ↔ (𝐹𝑥) = 𝐶))
109rexbidv 3256 . . . . . 6 (𝑦 = 𝐶 → (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
118, 10bibi12d 349 . . . . 5 (𝑦 = 𝐶 → ((𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦) ↔ (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)))
1211imbi2d 344 . . . 4 (𝑦 = 𝐶 → (((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))))
13 fnfun 6423 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
14 fndm 6425 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
1514sseq2d 3947 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
1615biimpar 481 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
17 dfimafn 6703 . . . . . 6 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1813, 16, 17syl2an2r 684 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1918abeq2d 2924 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
2012, 19vtoclg 3515 . . 3 (𝐶 ∈ V → ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)))
2120impcom 411 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
222, 7, 21pm5.21nd 801 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {cab 2776  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  dom cdm 5519   “ cima 5522  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-fv 6332 This theorem is referenced by:  fvelimabd  6713  unima  6714  ssimaex  6723  rexima  6977  ralima  6978  f1elima  6999  ovelimab  7306  fimaproj  7812  tcrank  9297  djuun  9339  ackbij2  9654  fin1a2lem6  9816  iunfo  9950  grothomex  10240  axpre-sup  10580  injresinjlem  13152  txkgen  22257  fmucndlem  22897  efopn  25249  pjimai  29959  fimarab  30404  qtophaus  31189  indf1ofs  31395  eulerpartgbij  31740  eulerpartlemgvv  31744  ballotlemsima  31883  elmthm  32936  elintfv  33120  nocvxmin  33361  isnacs2  39645  isnacs3  39649  islmodfg  40011  kercvrlsm  40025  isnumbasgrplem2  40046  dfacbasgrp  40050  fourierdlem62  42808
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