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Mirrors > Home > ILE Home > Th. List > cnvss | GIF version |
Description: Subset theorem for converse. (Contributed by NM, 22-Mar-1998.) |
Ref | Expression |
---|---|
cnvss | ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 2993 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (〈𝑦, 𝑥〉 ∈ 𝐴 → 〈𝑦, 𝑥〉 ∈ 𝐵)) | |
2 | df-br 3786 | . . . 4 ⊢ (𝑦𝐴𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝐴) | |
3 | df-br 3786 | . . . 4 ⊢ (𝑦𝐵𝑥 ↔ 〈𝑦, 𝑥〉 ∈ 𝐵) | |
4 | 1, 2, 3 | 3imtr4g 203 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑦𝐴𝑥 → 𝑦𝐵𝑥)) |
5 | 4 | ssopab2dv 4033 | . 2 ⊢ (𝐴 ⊆ 𝐵 → {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} ⊆ {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥}) |
6 | df-cnv 4371 | . 2 ⊢ ◡𝐴 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐴𝑥} | |
7 | df-cnv 4371 | . 2 ⊢ ◡𝐵 = {〈𝑥, 𝑦〉 ∣ 𝑦𝐵𝑥} | |
8 | 5, 6, 7 | 3sstr4g 3040 | 1 ⊢ (𝐴 ⊆ 𝐵 → ◡𝐴 ⊆ ◡𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1433 ⊆ wss 2973 〈cop 3401 class class class wbr 3785 {copab 3838 ◡ccnv 4362 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-in 2979 df-ss 2986 df-br 3786 df-opab 3840 df-cnv 4371 |
This theorem is referenced by: cnveq 4527 rnss 4582 relcnvtr 4860 funss 4940 funcnvuni 4988 funres11 4991 funcnvres 4992 foimacnv 5164 tposss 5884 |
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