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Theorem ixpssmap2g 6845
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 6846 avoids ax-rep 4131. (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 6838 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
21adantl 452 . . . 4  |-  ( (
U_ x  e.  A  B  e.  V  /\  f  e.  X_ x  e.  A  B )  -> 
f : A --> U_ x  e.  A  B )
3 n0i 3460 . . . . . 6  |-  ( f  e.  X_ x  e.  A  B  ->  -.  X_ x  e.  A  B  =  (/) )
4 ixpprc 6837 . . . . . 6  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
53, 4nsyl2 119 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
6 elmapg 6785 . . . . 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 460 . . . 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 223 . . 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 423 . 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 3185 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    C_ wss 3152   (/)c0 3455   U_ciun 3905   -->wf 5251  (class class class)co 5858    ^m cmap 6772   X_cixp 6817
This theorem is referenced by:  ixpssmapg  6846  ixpfi  7153  ixpiunwdom  7305  prdsval  13355  prdsbas  13357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774  df-ixp 6818
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