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Theorem rexsupp 6936
Description: Existential quantification restricted to a support. (Contributed by Stefan O'Rear, 23-Mar-2015.) (Revised by AV, 27-May-2019.)
Assertion
Ref Expression
rexsupp  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( E. x  e.  ( F supp  Z )
ph 
<->  E. x  e.  X  ( ( F `  x )  =/=  Z  /\  ph ) ) )
Distinct variable groups:    x, F    x, V    x, W    x, X    x, Z
Allowed substitution hint:    ph( x)

Proof of Theorem rexsupp
StepHypRef Expression
1 elsuppfn 6925 . . . 4  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( x  e.  ( F supp  Z )  <->  ( x  e.  X  /\  ( F `  x )  =/=  Z ) ) )
21anbi1d 704 . . 3  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( ( x  e.  ( F supp  Z )  /\  ph )  <->  ( (
x  e.  X  /\  ( F `  x )  =/=  Z )  /\  ph ) ) )
3 anass 649 . . 3  |-  ( ( ( x  e.  X  /\  ( F `  x
)  =/=  Z )  /\  ph )  <->  ( x  e.  X  /\  (
( F `  x
)  =/=  Z  /\  ph ) ) )
42, 3syl6bb 261 . 2  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( ( x  e.  ( F supp  Z )  /\  ph )  <->  ( x  e.  X  /\  (
( F `  x
)  =/=  Z  /\  ph ) ) ) )
54rexbidv2 2964 1  |-  ( ( F  Fn  X  /\  X  e.  V  /\  Z  e.  W )  ->  ( E. x  e.  ( F supp  Z )
ph 
<->  E. x  e.  X  ( ( F `  x )  =/=  Z  /\  ph ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1819    =/= wne 2652   E.wrex 2808    Fn wfn 5589   ` cfv 5594  (class class class)co 6296   supp csupp 6917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-supp 6918
This theorem is referenced by:  mdegldg  22592
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