Theorem cnvss 4886

Description: Subset theorem for converse. (Contributed by set.mm contributors, 22-Mar-1998.)

Assertion

Ref	Expression
cnvss	⊢ (A ⊆ B → ◡A ⊆ ◡B)

Proof of Theorem cnvss

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . 4 ⊢ (A ⊆ B → (⟨y, x⟩ ∈ A → ⟨y, x⟩ ∈ B))
2		df-br 4641	. . . 4 ⊢ (yAx ↔ ⟨y, x⟩ ∈ A)
3		df-br 4641	. . . 4 ⊢ (yBx ↔ ⟨y, x⟩ ∈ B)
4	1, 2, 3	3imtr4g 261	. . 3 ⊢ (A ⊆ B → (yAx → yBx))
5	4	ssopab2dv 4716	. 2 ⊢ (A ⊆ B → {⟨x, y⟩ ∣ yAx} ⊆ {⟨x, y⟩ ∣ yBx})
6		df-cnv 4786	. 2 ⊢ ◡A = {⟨x, y⟩ ∣ yAx}
7		df-cnv 4786	. 2 ⊢ ◡B = {⟨x, y⟩ ∣ yBx}
8	5, 6, 7	3sstr4g 3313	1 ⊢ (A ⊆ B → ◡A ⊆ ◡B)

Syntax hints:   → wi 4   ∈ wcel 1710   ⊆ wss 3258  ⟨cop 4562  {copab 4623   class class class wbr 4640  ^◡ccnv 4772

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-opab 4624  df-br 4641  df-cnv 4786

This theorem is referenced by:  cnveq  4887  rnss  4960  cnvtr  5099  funss  5127  funcnvuni  5162  funres11  5165  funcnvres  5166  foimacnv  5304