Theorem cnvss 5897

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 5210	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
2	1	ssopab2dv 5570	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
3		df-cnv 5708	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
4		df-cnv 5708	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
5	2, 3, 4	3sstr4g 4054	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

Syntax hints:   → wi 4   ⊆ wss 3976   class class class wbr 5166  {copab 5228  ^◡ccnv 5699

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ss 3993  df-br 5167  df-opab 5229  df-cnv 5708

This theorem is referenced by:  cnveq  5898  rnss  5964  relcnvtrg  6297  predrelss  6369  funss  6597  funres11  6655  funcnvres  6656  foimacnv  6879  funcnvuni  7972  tposss  8268  vdwnnlem1  17042  structcnvcnv  17200  catcoppccl  18184  catcoppcclOLD  18185  cnvps  18648  tsrdir  18674  ustneism  24253  metustsym  24589  metust  24592  pi1xfrcnv  25109  eulerpartlemmf  34340  relcnveq3  38277  elrelscnveq3  38447  disjss  38687  cnvssb  43548  trclubgNEW  43580  clrellem  43584  clcnvlem  43585  cnvrcl0  43587  cnvtrcl0  43588  cnvtrrel  43632  relexpaddss  43680