Theorem ssres2 2590

Description: Subclass theorem for restriction.

Assertion

Ref	Expression
ssres2	⊢ (A ⊆ B → (C ↾ A) ⊆ (C ↾ B))

Proof of Theorem ssres2

Step	Hyp	Ref	Expression
1		ssid 1519	. . . 4 ⊢ V ⊆ V
2		ssxp 2487	. . . 4 ⊢ ((A ⊆ B ∧ V ⊆ V) → (A × V) ⊆ (B × V))
3	1, 2	mpan2 519	. . 3 ⊢ (A ⊆ B → (A × V) ⊆ (B × V))
4		sslin 1662	. . 3 ⊢ ((A × V) ⊆ (B × V) → (C ∩ (A × V)) ⊆ (C ∩ (B × V)))
5	3, 4	syl 12	. 2 ⊢ (A ⊆ B → (C ∩ (A × V)) ⊆ (C ∩ (B × V)))
6		df-res 2430	. 2 ⊢ (C ↾ A) = (C ∩ (A × V))
7		df-res 2430	. 2 ⊢ (C ↾ B) = (C ∩ (B × V))
8	5, 6, 7	3sstr4g 1541	1 ⊢ (A ⊆ B → (C ↾ A) ⊆ (C ↾ B))

Syntax hints:   → wi 2  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487   × cxp 2408   ↾ cres 2412

This theorem is referenced by:  imass2 2622

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-res 2430

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