MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ss2ixp Structured version   Visualization version   GIF version

Theorem ss2ixp 7865
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 3577 . . . . 5 (𝐵𝐶 → ((𝑓𝑥) ∈ 𝐵 → (𝑓𝑥) ∈ 𝐶))
21ral2imi 2942 . . . 4 (∀𝑥𝐴 𝐵𝐶 → (∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵 → ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶))
32anim2d 588 . . 3 (∀𝑥𝐴 𝐵𝐶 → ((𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵) → (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)))
43ss2abdv 3654 . 2 (∀𝑥𝐴 𝐵𝐶 → {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)} ⊆ {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)})
5 df-ixp 7853 . 2 X𝑥𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐵)}
6 df-ixp 7853 . 2 X𝑥𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥𝑥𝐴} ∧ ∀𝑥𝐴 (𝑓𝑥) ∈ 𝐶)}
74, 5, 63sstr4g 3625 1 (∀𝑥𝐴 𝐵𝐶X𝑥𝐴 𝐵X𝑥𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  {cab 2607  wral 2907  wss 3555   Fn wfn 5842  cfv 5847  Xcixp 7852
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-in 3562  df-ss 3569  df-ixp 7853
This theorem is referenced by:  ixpeq2  7866  boxcutc  7895  pwcfsdom  9349  prdsval  16036  prdshom  16048  sscpwex  16396  wunfunc  16480  wunnat  16537  dprdss  18349  psrbaglefi  19291  ptuni2  21289  ptcld  21326  ptclsg  21328  prdstopn  21341  xkopt  21368  tmdgsum2  21810  ressprdsds  22086  prdsbl  22206  ptrecube  33041  prdstotbnd  33225  ixpssixp  38754  ioorrnopnxrlem  39833  ovnlecvr2  40131
  Copyright terms: Public domain W3C validator