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Theorem ixpssmap2g 7556
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7557 avoids ax-rep 4518. (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 7549 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f : A --> U_ x  e.  A  B
)
21adantl 468 . . . 4  |-  ( (
U_ x  e.  A  B  e.  V  /\  f  e.  X_ x  e.  A  B )  -> 
f : A --> U_ x  e.  A  B )
3 n0i 3738 . . . . . 6  |-  ( f  e.  X_ x  e.  A  B  ->  -.  X_ x  e.  A  B  =  (/) )
4 ixpprc 7548 . . . . . 6  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
53, 4nsyl2 131 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
6 elmapg 7490 . . . . 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 477 . . . 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 236 . . 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 436 . 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 3440 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 188    /\ wa 371    = wceq 1446    e. wcel 1889   _Vcvv 3047    C_ wss 3406   (/)c0 3733   U_ciun 4281   -->wf 5581  (class class class)co 6295    ^m cmap 7477   X_cixp 7527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7479  df-ixp 7528
This theorem is referenced by:  ixpssmapg  7557  ixpfi  7876  ixpiunwdom  8111  prdsval  15365  prdsbas  15367  ixpssmapc  37428
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