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Theorem ixpssmap2g 8985
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8986 avoids ax-rep 5303. (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 8978 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
21adantl 481 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓:𝐴 𝑥𝐴 𝐵)
3 n0i 4363 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ¬ X𝑥𝐴 𝐵 = ∅)
4 ixpprc 8977 . . . . . 6 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
53, 4nsyl2 141 . . . . 5 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
6 elmapg 8897 . . . . 5 (( 𝑥𝐴 𝐵𝑉𝐴 ∈ V) → (𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴) ↔ 𝑓:𝐴 𝑥𝐴 𝐵))
75, 6sylan2 592 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → (𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴) ↔ 𝑓:𝐴 𝑥𝐴 𝐵))
82, 7mpbird 257 . . 3 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴))
98ex 412 . 2 ( 𝑥𝐴 𝐵𝑉 → (𝑓X𝑥𝐴 𝐵𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴)))
109ssrdv 4014 1 ( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2108  Vcvv 3488  wss 3976  c0 4352   ciun 5015  wf 6569  (class class class)co 7448  m cmap 8884  Xcixp 8955
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-ixp 8956
This theorem is referenced by:  ixpssmapg  8986  ixpfi  9419  ixpiunwdom  9659  prdsval  17515  prdsbas  17517  ixpssmapc  44975
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