Theorem cnvss 5871

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 5192	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
2	1	ssopab2dv 5551	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
3		df-cnv 5684	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
4		df-cnv 5684	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
5	2, 3, 4	3sstr4g 4027	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Syntax hints:   → wi 4   ⊆ wss 3948   class class class wbr 5148  {copab 5210  ^◡ccnv 5675

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 3955  df-ss 3965  df-br 5149  df-opab 5211  df-cnv 5684

This theorem is referenced by:  cnveq  5872  rnss  5937  relcnvtrg  6263  predrelss  6336  funss  6565  funres11  6623  funcnvres  6624  foimacnv  6848  funcnvuni  7919  tposss  8209  vdwnnlem1  16925  structcnvcnv  17083  catcoppccl  18064  catcoppcclOLD  18065  cnvps  18528  tsrdir  18554  ustneism  23720  metustsym  24056  metust  24059  pi1xfrcnv  24565  eulerpartlemmf  33363  relcnveq3  37179  elrelscnveq3  37350  disjss  37590  cnvssb  42323  trclubgNEW  42355  clrellem  42359  clcnvlem  42360  cnvrcl0  42362  cnvtrcl0  42363  cnvtrrel  42407  relexpaddss  42455