Theorem cnvss 5856

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 5152	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥))
2	1	ssopab2dv 5530	. 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥})
3		df-cnv 5663	. 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥}
4		df-cnv 5663	. 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}
5	2, 3, 4	3sstr4g 3966	1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   class class class wbr 5106  {copab 5170  ^◡ccnv 5654

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-br 5107  df-opab 5171  df-cnv 5663

This theorem is used by:  cnveq  5857  rnss  5927  cnvssb  6200  relcnvtrg  6283  relcnvtrgOLD  6284  predrelss  6355  funss  6575  funres11  6636  funcnvres  6637  foimacnv  6861  funcnvuni  7965  tposss  8269  vdwnnlem1  17199  structcnvcnv  17357  catcoppccl  18318  cnvps  18778  tsrdir  18804  ustneism  24570  metustsym  24901  metust  24904  pi1xfrcnv  25405  eulerpartlemmf  35074  relcnveq3  39450  elrelscnveq3  39750  disjss  39954  trclubgNEW  44808  clrellem  44812  clcnvlem  44813  cnvrcl0  44815  cnvtrcl0  44816  cnvtrrel  44860  relexpaddss  44908