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Theorem ixpssmap2g 8868
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8869 avoids ax-rep 5243. (Contributed by Mario Carneiro, 16-Nov-2014.)
Assertion
Ref Expression
ixpssmap2g ( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 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 8861 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
21adantl 483 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓:𝐴 𝑥𝐴 𝐵)
3 n0i 4294 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ¬ X𝑥𝐴 𝐵 = ∅)
4 ixpprc 8860 . . . . . 6 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
53, 4nsyl2 141 . . . . 5 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
6 elmapg 8781 . . . . 5 (( 𝑥𝐴 𝐵𝑉𝐴 ∈ V) → (𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴) ↔ 𝑓:𝐴 𝑥𝐴 𝐵))
75, 6sylan2 594 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → (𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴) ↔ 𝑓:𝐴 𝑥𝐴 𝐵))
82, 7mpbird 257 . . 3 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴))
98ex 414 . 2 ( 𝑥𝐴 𝐵𝑉 → (𝑓X𝑥𝐴 𝐵𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴)))
109ssrdv 3951 1 ( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  Vcvv 3444  wss 3911  c0 4283   ciun 4955  wf 6493  (class class class)co 7358  m cmap 8768  Xcixp 8838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-ixp 8839
This theorem is referenced by:  ixpssmapg  8869  ixpfi  9296  ixpiunwdom  9531  prdsval  17342  prdsbas  17344  ixpssmapc  43370
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