Theorem ss2ixp 8939

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 3902	. . . . 5 ⊢ (𝐵 ⊆ 𝐶 → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ 𝐶))
2	1	ral2imi 3101	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶))
3	2	anim2d 624	. . 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 3990	. 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 8927	. 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)}
6		df-ixp 8927	. 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)}
7	4, 5, 6	3sstr4g 3961	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144  {cab 2738  ∀wral 3076   ⊆ wss 3895   Fn wfn 6533  ‘cfv 6538  Xcixp 8926

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-ss 3896  df-ixp 8927

This theorem is used by:  ixpeq2  8940  boxcutc  8970  pwcfsdom  10669  prdsvallem  17626  prdshom  17639  sscpwex  17991  wunfunc  18077  wunnat  18135  dprdss  20246  psrbaglefi  22235  ptuni2  23896  ptcld  23933  ptclsg  23935  prdstopn  23948  xkopt  23975  tmdgsum2  24416  ressprdsds  24691  prdsbl  24811  ptrecube  38614  prdstotbnd  38804  ixpssixp  46183  ioorrnopnxrlem  47392  ovnlecvr2  47696