Theorem cnvss 5873

Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.)

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		ssbr 5193	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
2	1	ssopab2dv 5552	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
3		df-cnv 5685	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
4		df-cnv 5685	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
5	2, 3, 4	3sstr4g 4028	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Syntax hints:   → wi 4   ⊆ wss 3949   class class class wbr 5149  {copab 5211  ^◡ccnv 5676

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-br 5150  df-opab 5212  df-cnv 5685

This theorem is referenced by:  cnveq  5874  rnss  5939  relcnvtrg  6266  predrelss  6339  funss  6568  funres11  6626  funcnvres  6627  foimacnv  6851  funcnvuni  7922  tposss  8212  vdwnnlem1  16928  structcnvcnv  17086  catcoppccl  18067  catcoppcclOLD  18068  cnvps  18531  tsrdir  18557  ustneism  23728  metustsym  24064  metust  24067  pi1xfrcnv  24573  eulerpartlemmf  33374  relcnveq3  37190  elrelscnveq3  37361  disjss  37601  cnvssb  42337  trclubgNEW  42369  clrellem  42373  clcnvlem  42374  cnvrcl0  42376  cnvtrcl0  42377  cnvtrrel  42421  relexpaddss  42469