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Theorem fvelimab 6215
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 3201 . . 3 (𝐶 ∈ (𝐹𝐵) → 𝐶 ∈ V)
21anim2i 592 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ (𝐹𝐵)) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
3 fvex 6163 . . . . 5 (𝐹𝑥) ∈ V
4 eleq1 2686 . . . . 5 ((𝐹𝑥) = 𝐶 → ((𝐹𝑥) ∈ V ↔ 𝐶 ∈ V))
53, 4mpbii 223 . . . 4 ((𝐹𝑥) = 𝐶𝐶 ∈ V)
65rexlimivw 3023 . . 3 (∃𝑥𝐵 (𝐹𝑥) = 𝐶𝐶 ∈ V)
76anim2i 592 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ ∃𝑥𝐵 (𝐹𝑥) = 𝐶) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
8 eleq1 2686 . . . . . 6 (𝑦 = 𝐶 → (𝑦 ∈ (𝐹𝐵) ↔ 𝐶 ∈ (𝐹𝐵)))
9 eqeq2 2632 . . . . . . 7 (𝑦 = 𝐶 → ((𝐹𝑥) = 𝑦 ↔ (𝐹𝑥) = 𝐶))
109rexbidv 3046 . . . . . 6 (𝑦 = 𝐶 → (∃𝑥𝐵 (𝐹𝑥) = 𝑦 ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
118, 10bibi12d 335 . . . . 5 (𝑦 = 𝐶 → ((𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦) ↔ (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)))
1211imbi2d 330 . . . 4 (𝑦 = 𝐶 → (((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))))
13 fnfun 5951 . . . . . 6 (𝐹 Fn 𝐴 → Fun 𝐹)
14 fndm 5953 . . . . . . . 8 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
1514sseq2d 3617 . . . . . . 7 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
1615biimpar 502 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
17 dfimafn 6207 . . . . . 6 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1813, 16, 17syl2an2r 875 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) = {𝑦 ∣ ∃𝑥𝐵 (𝐹𝑥) = 𝑦})
1918abeq2d 2731 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑦 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝑦))
2012, 19vtoclg 3255 . . 3 (𝐶 ∈ V → ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)))
2120impcom 446 . 2 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
222, 7, 21pm5.21nd 940 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  Vcvv 3189  wss 3559  dom cdm 5079  cima 5082  Fun wfun 5846   Fn wfn 5847  cfv 5852
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-fv 5860
This theorem is referenced by:  fvelimabd  6216  ssimaex  6225  rexima  6457  ralima  6458  f1elima  6480  ovelimab  6772  tcrank  8699  ackbij2  9017  fin1a2lem6  9179  iunfo  9313  grothomex  9603  axpre-sup  9942  injresinjlem  12536  lmhmima  18979  txkgen  21378  fmucndlem  22018  mdegldg  23747  ig1peu  23852  efopn  24321  pjimai  28905  fimarab  29310  fimaproj  29706  qtophaus  29709  indf1ofs  29893  eulerpartgbij  30239  eulerpartlemgvv  30243  ballotlemsima  30382  elmthm  31216  nocvxmin  31589  isnacs2  36784  isnacs3  36788  islmodfg  37154  kercvrlsm  37168  isnumbasgrplem2  37190  dfacbasgrp  37194  unima  38851  fourierdlem62  39718
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