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Theorem ixpssmap 8909
Description: An infinite Cartesian product is a subset of set exponentiation. Remark in [Enderton] p. 54. (Contributed by NM, 28-Sep-2006.)
Hypothesis
Ref Expression
ixpssmap.2 𝐵 ∈ V
Assertion
Ref Expression
ixpssmap X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ixpssmap
StepHypRef Expression
1 ixpssmap.2 . . 3 𝐵 ∈ V
21rgenw 3064 . 2 𝑥𝐴 𝐵 ∈ V
3 ixpssmapg 8905 . 2 (∀𝑥𝐴 𝐵 ∈ V → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
42, 3ax-mp 5 1 X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  wral 3060  Vcvv 3473  wss 3944   ciun 4990  (class class class)co 7393  m cmap 8803  Xcixp 8874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-map 8805  df-ixp 8875
This theorem is referenced by:  hspmbl  45116
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