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Theorem ixpssmap2g 8479
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8480 avoids ax-rep 5181. (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 8472 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
21adantl 482 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓:𝐴 𝑥𝐴 𝐵)
3 n0i 4296 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ¬ X𝑥𝐴 𝐵 = ∅)
4 ixpprc 8471 . . . . . 6 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
53, 4nsyl2 143 . . . . 5 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
6 elmapg 8408 . . . . 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 258 . . 3 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴))
98ex 413 . 2 ( 𝑥𝐴 𝐵𝑉 → (𝑓X𝑥𝐴 𝐵𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴)))
109ssrdv 3970 1 ( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  wss 3933  c0 4288   ciun 4910  wf 6344  (class class class)co 7145  m cmap 8395  Xcixp 8449
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-ixp 8450
This theorem is referenced by:  ixpssmapg  8480  ixpfi  8809  ixpiunwdom  9043  prdsval  16716  prdsbas  16718  ixpssmapc  41213
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