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Theorem ixpssmap2g 7091
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7092 avoids ax-rep 4320. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
ixpssmap2g  |-  ( U_ x  e.  A  B  e.  V  ->  X_ x  e.  A  B  C_  ( U_ x  e.  A  B  ^m  A ) )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    V( x)

Proof of Theorem ixpssmap2g
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 ixpf 7084 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
21adantl 453 . . . 4  |-  ( (
U_ x  e.  A  B  e.  V  /\  f  e.  X_ x  e.  A  B )  -> 
f : A --> U_ x  e.  A  B )
3 n0i 3633 . . . . . 6  |-  ( f  e.  X_ x  e.  A  B  ->  -.  X_ x  e.  A  B  =  (/) )
4 ixpprc 7083 . . . . . 6  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
53, 4nsyl2 121 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
6 elmapg 7031 . . . . 5  |-  ( (
U_ x  e.  A  B  e.  V  /\  A  e.  _V )  ->  ( f  e.  (
U_ x  e.  A  B  ^m  A )  <->  f : A
--> U_ x  e.  A  B ) )
75, 6sylan2 461 . . . 4  |-  ( (
U_ x  e.  A  B  e.  V  /\  f  e.  X_ x  e.  A  B )  -> 
( f  e.  (
U_ x  e.  A  B  ^m  A )  <->  f : A
--> U_ x  e.  A  B ) )
82, 7mpbird 224 . . 3  |-  ( (
U_ x  e.  A  B  e.  V  /\  f  e.  X_ x  e.  A  B )  -> 
f  e.  ( U_ x  e.  A  B  ^m  A ) )
98ex 424 . 2  |-  ( U_ x  e.  A  B  e.  V  ->  ( f  e.  X_ x  e.  A  B  ->  f  e.  (
U_ x  e.  A  B  ^m  A ) ) )
109ssrdv 3354 1  |-  ( U_ x  e.  A  B  e.  V  ->  X_ x  e.  A  B  C_  ( U_ x  e.  A  B  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    C_ wss 3320   (/)c0 3628   U_ciun 4093   -->wf 5450  (class class class)co 6081    ^m cmap 7018   X_cixp 7063
This theorem is referenced by:  ixpssmapg  7092  ixpfi  7404  ixpiunwdom  7559  prdsval  13678  prdsbas  13680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-ixp 7064
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