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Theorem ixpssmap2g 7496
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7497 avoids ax-rep 4544. (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 7489 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
21adantl 466 . . . 4  |-  ( (
U_ x  e.  A  B  e.  V  /\  f  e.  X_ x  e.  A  B )  -> 
f : A --> U_ x  e.  A  B )
3 n0i 3772 . . . . . 6  |-  ( f  e.  X_ x  e.  A  B  ->  -.  X_ x  e.  A  B  =  (/) )
4 ixpprc 7488 . . . . . 6  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
53, 4nsyl2 127 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
6 elmapg 7431 . . . . 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 474 . . . 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 232 . . 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 434 . 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 3492 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   _Vcvv 3093    C_ wss 3458   (/)c0 3767   U_ciun 4311   -->wf 5570  (class class class)co 6277    ^m cmap 7418   X_cixp 7467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7420  df-ixp 7468
This theorem is referenced by:  ixpssmapg  7497  ixpfi  7815  ixpiunwdom  8015  prdsval  14724  prdsbas  14726
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