Theorem ss2ixp 8903

Description: Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.)

Assertion

Ref	Expression
ss2ixp	⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)

Proof of Theorem ss2ixp

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3975	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ 𝐶))
2	1	ral2imi 3085	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
3	2	anim2d 612	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ((𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵) → (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)))
4	3	ss2abdv 4060	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} ⊆ {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)})
5		df-ixp 8891	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
6		df-ixp 8891	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)}
7	4, 5, 6	3sstr4g 4027	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 396   ∈ wcel 2106  {cab 2709  ∀wral 3061   ⊆ wss 3948   Fn wfn 6538  ‘cfv 6543  Xcixp 8890

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-v 3476  df-in 3955  df-ss 3965  df-ixp 8891

This theorem is referenced by:  ixpeq2  8904  boxcutc  8934  pwcfsdom  10577  prdsvallem  17399  prdshom  17412  sscpwex  17761  wunfunc  17848  wunfuncOLD  17849  wunnat  17906  wunnatOLD  17907  dprdss  19898  psrbaglefi  21484  psrbaglefiOLD  21485  ptuni2  23079  ptcld  23116  ptclsg  23118  prdstopn  23131  xkopt  23158  tmdgsum2  23599  ressprdsds  23876  prdsbl  23999  ptrecube  36483  prdstotbnd  36657  ixpssixp  43771  ioorrnopnxrlem  45012  ovnlecvr2  45316