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Theorem ixpssmap2g 7934
Description: An infinite Cartesian product is a subset of set exponentiation. This version of ixpssmapg 7935 avoids ax-rep 4769. (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 7927 . . . . 5 (𝑓X𝑥𝐴 𝐵𝑓:𝐴 𝑥𝐴 𝐵)
21adantl 482 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓:𝐴 𝑥𝐴 𝐵)
3 n0i 3918 . . . . . 6 (𝑓X𝑥𝐴 𝐵 → ¬ X𝑥𝐴 𝐵 = ∅)
4 ixpprc 7926 . . . . . 6 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
53, 4nsyl2 142 . . . . 5 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
6 elmapg 7867 . . . . 5 (( 𝑥𝐴 𝐵𝑉𝐴 ∈ V) → (𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 𝐴) ↔ 𝑓:𝐴 𝑥𝐴 𝐵))
75, 6sylan2 491 . . . 4 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → (𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 𝐴) ↔ 𝑓:𝐴 𝑥𝐴 𝐵))
82, 7mpbird 247 . . 3 (( 𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 𝐴))
98ex 450 . 2 ( 𝑥𝐴 𝐵𝑉 → (𝑓X𝑥𝐴 𝐵𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 𝐴)))
109ssrdv 3607 1 ( 𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  wss 3572  c0 3913   ciun 4518  wf 5882  (class class class)co 6647  𝑚 cmap 7854  Xcixp 7905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-ov 6650  df-oprab 6651  df-mpt2 6652  df-map 7856  df-ixp 7906
This theorem is referenced by:  ixpssmapg  7935  ixpfi  8260  ixpiunwdom  8493  prdsval  16109  prdsbas  16111  ixpssmapc  39069
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