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Mirrors > Home > MPE Home > Th. List > ss2ixp | Structured version Visualization version GIF version |
Description: Subclass theorem for infinite Cartesian product. (Contributed by NM, 29-Sep-2006.) (Revised by Mario Carneiro, 12-Aug-2016.) |
Ref | Expression |
---|---|
ss2ixp | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3562 | . . . . 5 ⊢ (𝐵 ⊆ 𝐶 → ((𝑓‘𝑥) ∈ 𝐵 → (𝑓‘𝑥) ∈ 𝐶)) | |
2 | 1 | ral2imi 2931 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → (∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵 → ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)) |
3 | 2 | anim2d 587 | . . 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 3638 | . 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 7795 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐵)} | |
6 | df-ixp 7795 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐶 = {𝑓 ∣ (𝑓 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑓‘𝑥) ∈ 𝐶)} | |
7 | 4, 5, 6 | 3sstr4g 3609 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → X𝑥 ∈ 𝐴 𝐵 ⊆ X𝑥 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ wcel 1977 {cab 2596 ∀wral 2896 ⊆ wss 3540 Fn wfn 5799 ‘cfv 5804 Xcixp 7794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-in 3547 df-ss 3554 df-ixp 7795 |
This theorem is referenced by: ixpeq2 7808 boxcutc 7837 pwcfsdom 9284 prdsval 15938 prdshom 15950 sscpwex 16298 wunfunc 16382 wunnat 16439 dprdss 18251 psrbaglefi 19193 ptuni2 21189 ptcld 21226 ptclsg 21228 prdstopn 21241 xkopt 21268 tmdgsum2 21710 ressprdsds 21986 prdsbl 22106 ptrecube 32579 prdstotbnd 32763 ixpssixp 38297 ioorrnopnxrlem 39202 ovnlecvr2 39500 |
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