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Theorem ixpssmap2g 8466
 Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 8467 avoids ax-rep 5163. (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 8459 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
21adantl 485 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓:𝐴 𝑥𝐴 𝐵)
3 n0i 4272 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ¬ X𝑥𝐴 𝐵 = ∅)
4 ixpprc 8458 . . . . . 6 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
53, 4nsyl2 143 . . . . 5 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
6 elmapg 8394 . . . . 5 (( 𝑥𝐴 𝐵𝑉𝐴 ∈ V) → (𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴) ↔ 𝑓:𝐴 𝑥𝐴 𝐵))
75, 6sylan2 595 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → (𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴) ↔ 𝑓:𝐴 𝑥𝐴 𝐵))
82, 7mpbird 260 . . 3 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴))
98ex 416 . 2 ( 𝑥𝐴 𝐵𝑉 → (𝑓X𝑥𝐴 𝐵𝑓 ∈ ( 𝑥𝐴 𝐵m 𝐴)))
109ssrdv 3949 1 ( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵m 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  Vcvv 3471   ⊆ wss 3910  ∅c0 4266  ∪ ciun 4892  ⟶wf 6324  (class class class)co 7130   ↑m cmap 8381  Xcixp 8436 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-map 8383  df-ixp 8437 This theorem is referenced by:  ixpssmapg  8467  ixpfi  8797  ixpiunwdom  9030  prdsval  16706  prdsbas  16708  ixpssmapc  41493
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