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Theorem fvelimab 6568
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 3433 . . 3 (𝐶 ∈ (𝐹𝐵) → 𝐶 ∈ V)
21anim2i 607 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ (𝐹𝐵)) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
3 fvex 6514 . . . . 5 (𝐹𝑥) ∈ V
4 eleq1 2853 . . . . 5 ((𝐹𝑥) = 𝐶 → ((𝐹𝑥) ∈ V ↔ 𝐶 ∈ V))
53, 4mpbii 225 . . . 4 ((𝐹𝑥) = 𝐶𝐶 ∈ V)
65rexlimivw 3227 . . 3 (∃𝑥𝐵 (𝐹𝑥) = 𝐶𝐶 ∈ V)
76anim2i 607 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ ∃𝑥𝐵 (𝐹𝑥) = 𝐶) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
8 eleq1 2853 . . . . . 6 (𝑦 = 𝐶 → (𝑦 ∈ (𝐹𝐵) ↔ 𝐶 ∈ (𝐹𝐵)))
9 eqeq2 2789 . . . . . . 7 (𝑦 = 𝐶 → ((𝐹𝑥) = 𝑦 ↔ (𝐹𝑥) = 𝐶))
109rexbidv 3242 . . . . . 6 (𝑦 = 𝐶 → (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
118, 10bibi12d 338 . . . . 5 (𝑦 = 𝐶 → ((𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦) ↔ (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)))
1211imbi2d 333 . . . 4 (𝑦 = 𝐶 → (((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))))
13 fnfun 6288 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
14 fndm 6290 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
1514sseq2d 3891 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
1615biimpar 470 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
17 dfimafn 6560 . . . . . 6 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1813, 16, 17syl2an2r 672 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1918abeq2d 2899 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
2012, 19vtoclg 3486 . . 3 (𝐶 ∈ V → ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)))
2120impcom 399 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
222, 7, 21pm5.21nd 789 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  {cab 2758  wrex 3089  Vcvv 3415  wss 3831  dom cdm 5408  cima 5411  Fun wfun 6184   Fn wfn 6185  cfv 6190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fn 6193  df-fv 6198
This theorem is referenced by:  fvelimabd  6569  ssimaex  6578  rexima  6826  ralima  6827  f1elima  6848  ovelimab  7144  tcrank  9109  djuun  9151  ackbij2  9465  fin1a2lem6  9627  iunfo  9761  grothomex  10051  axpre-sup  10391  injresinjlem  12975  txkgen  21967  fmucndlem  22606  efopn  24945  pjimai  29737  fimarab  30155  fimaproj  30741  qtophaus  30744  indf1ofs  30929  eulerpartgbij  31275  eulerpartlemgvv  31279  ballotlemsima  31419  elmthm  32343  elintfv  32527  nocvxmin  32769  isnacs2  38698  isnacs3  38702  islmodfg  39065  kercvrlsm  39079  isnumbasgrplem2  39100  dfacbasgrp  39104  unima  40847  fourierdlem62  41885
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