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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 5192 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥)) | |
2 | 1 | ssopab2dv 5561 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
3 | df-cnv 5697 | . 2 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
4 | df-cnv 5697 | . 2 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
5 | 2, 3, 4 | 3sstr4g 4041 | 1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3963 class class class wbr 5148 {copab 5210 ◡ccnv 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ss 3980 df-br 5149 df-opab 5211 df-cnv 5697 |
This theorem is referenced by: cnveq 5887 rnss 5953 relcnvtrg 6288 predrelss 6360 funss 6587 funres11 6645 funcnvres 6646 foimacnv 6866 funcnvuni 7955 tposss 8251 vdwnnlem1 17029 structcnvcnv 17187 catcoppccl 18171 catcoppcclOLD 18172 cnvps 18636 tsrdir 18662 ustneism 24248 metustsym 24584 metust 24587 pi1xfrcnv 25104 eulerpartlemmf 34357 relcnveq3 38303 elrelscnveq3 38473 disjss 38713 cnvssb 43576 trclubgNEW 43608 clrellem 43612 clcnvlem 43613 cnvrcl0 43615 cnvtrcl0 43616 cnvtrrel 43660 relexpaddss 43708 |
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