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Theorem ixpssmapg 7898
Description: An infinite Cartesian product is a subset of set exponentiation. (Contributed by Jeff Madsen, 19-Jun-2011.)
Assertion
Ref Expression
ixpssmapg (∀𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem ixpssmapg
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 n0i 3902 . . . . . . 7 (𝑓X𝑥𝐴 𝐵 → ¬ X𝑥𝐴 𝐵 = ∅)
2 ixpprc 7889 . . . . . . 7 𝐴 ∈ V → X𝑥𝐴 𝐵 = ∅)
31, 2nsyl2 142 . . . . . 6 (𝑓X𝑥𝐴 𝐵𝐴 ∈ V)
4 id 22 . . . . . 6 (∀𝑥𝐴 𝐵𝑉 → ∀𝑥𝐴 𝐵𝑉)
5 iunexg 7104 . . . . . 6 ((𝐴 ∈ V ∧ ∀𝑥𝐴 𝐵𝑉) → 𝑥𝐴 𝐵 ∈ V)
63, 4, 5syl2anr 495 . . . . 5 ((∀𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑥𝐴 𝐵 ∈ V)
7 ixpssmap2g 7897 . . . . 5 ( 𝑥𝐴 𝐵 ∈ V → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
86, 7syl 17 . . . 4 ((∀𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
9 simpr 477 . . . 4 ((∀𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓X𝑥𝐴 𝐵)
108, 9sseldd 3589 . . 3 ((∀𝑥𝐴 𝐵𝑉𝑓X𝑥𝐴 𝐵) → 𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 𝐴))
1110ex 450 . 2 (∀𝑥𝐴 𝐵𝑉 → (𝑓X𝑥𝐴 𝐵𝑓 ∈ ( 𝑥𝐴 𝐵𝑚 𝐴)))
1211ssrdv 3594 1 (∀𝑥𝐴 𝐵𝑉X𝑥𝐴 𝐵 ⊆ ( 𝑥𝐴 𝐵𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  wss 3560  c0 3897   ciun 4492  (class class class)co 6615  𝑚 cmap 7817  Xcixp 7868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819  df-ixp 7869
This theorem is referenced by:  ixpssmap  7902  gruixp  9591  hoissrrn  40100  hoissrrn2  40129
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