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Theorem ss2ixp 8477
Description: Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)
Assertion
Ref Expression
ss2ixp (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)

Proof of Theorem ss2ixp
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ssel 3964 . . . . 5 (𝐵𝐶 → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ 𝐶))
21ral2imi 3159 . . . 4 (∀𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
32anim2d 613 . . 3 (∀𝑥𝐴 𝐵𝐶 → ((𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) → (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
43ss2abdv 4047 . 2 (∀𝑥𝐴 𝐵𝐶 → {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)} ⊆ {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)})
5 df-ixp 8465 . 2 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
6 df-ixp 8465 . 2 X𝑥𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)}
74, 5, 63sstr4g 4015 1 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2113  {cab 2802  wral 3141  wss 3939   Fn wfn 6353  cfv 6358  Xcixp 8464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-in 3946  df-ss 3955  df-ixp 8465
This theorem is referenced by:  ixpeq2  8478  boxcutc  8508  pwcfsdom  10008  prdsval  16731  prdshom  16743  sscpwex  17088  wunfunc  17172  wunnat  17229  dprdss  19154  psrbaglefi  20155  ptuni2  22187  ptcld  22224  ptclsg  22226  prdstopn  22239  xkopt  22266  tmdgsum2  22707  ressprdsds  22984  prdsbl  23104  ptrecube  34896  prdstotbnd  35076  ixpssixp  41364  ioorrnopnxrlem  42598  ovnlecvr2  42899
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