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Mirrors > Home > MPE Home > Th. List > cnvss | Structured version Visualization version GIF version |
Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Kyle Wyonch, 27-Apr-2021.) |
Ref | Expression |
---|---|
cnvss | ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssbr 5191 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥)) | |
2 | 1 | ssopab2dv 5550 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥}) |
3 | df-cnv 5683 | . 2 ⊢ ◡𝐴 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐴𝑥} | |
4 | df-cnv 5683 | . 2 ⊢ ◡𝐵 = {⟨𝑥, 𝑦⟩ ∣ 𝑦𝐵𝑥} | |
5 | 2, 3, 4 | 3sstr4g 4026 | 1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3947 class class class wbr 5147 {copab 5209 ◡ccnv 5674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-v 3476 df-in 3954 df-ss 3964 df-br 5148 df-opab 5210 df-cnv 5683 |
This theorem is referenced by: cnveq 5871 rnss 5936 relcnvtrg 6262 predrelss 6335 funss 6564 funres11 6622 funcnvres 6623 foimacnv 6847 funcnvuni 7918 tposss 8208 vdwnnlem1 16924 structcnvcnv 17082 catcoppccl 18063 catcoppcclOLD 18064 cnvps 18527 tsrdir 18553 ustneism 23719 metustsym 24055 metust 24058 pi1xfrcnv 24564 eulerpartlemmf 33362 relcnveq3 37178 elrelscnveq3 37349 disjss 37589 cnvssb 42322 trclubgNEW 42354 clrellem 42358 clcnvlem 42359 cnvrcl0 42361 cnvtrcl0 42362 cnvtrrel 42406 relexpaddss 42454 |
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