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Mirrors > Home > MPE Home > Th. List > ssres2 | Structured version Visualization version GIF version |
Description: Subclass theorem for restriction. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
ssres2 | ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpss1 5228 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 × V) ⊆ (𝐵 × V)) | |
2 | sslin 3839 | . . 3 ⊢ ((𝐴 × V) ⊆ (𝐵 × V) → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V))) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ (𝐴 × V)) ⊆ (𝐶 ∩ (𝐵 × V))) |
4 | df-res 5126 | . 2 ⊢ (𝐶 ↾ 𝐴) = (𝐶 ∩ (𝐴 × V)) | |
5 | df-res 5126 | . 2 ⊢ (𝐶 ↾ 𝐵) = (𝐶 ∩ (𝐵 × V)) | |
6 | 3, 4, 5 | 3sstr4g 3646 | 1 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ↾ 𝐴) ⊆ (𝐶 ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Vcvv 3200 ∩ cin 3573 ⊆ wss 3574 × cxp 5112 ↾ cres 5116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-ss 3588 df-opab 4713 df-xp 5120 df-res 5126 |
This theorem is referenced by: imass2 5501 1stcof 7196 2ndcof 7197 tfrlem15 7488 gsum2dlem2 18370 txkgen 21455 funpsstri 31663 resnonrel 37898 mptrcllem 37920 rtrclexi 37928 cnvrcl0 37932 relexpss1d 37997 relexp0a 38008 supcnvlimsup 39972 |
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