New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > cnvss | GIF version |
Description: Subset theorem for converse. (Contributed by set.mm contributors, 22-Mar-1998.) |
Ref | Expression |
---|---|
cnvss | ⊢ (A ⊆ B → ◡A ⊆ ◡B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3267 | . . . 4 ⊢ (A ⊆ B → (〈y, x〉 ∈ A → 〈y, x〉 ∈ B)) | |
2 | df-br 4640 | . . . 4 ⊢ (yAx ↔ 〈y, x〉 ∈ A) | |
3 | df-br 4640 | . . . 4 ⊢ (yBx ↔ 〈y, x〉 ∈ B) | |
4 | 1, 2, 3 | 3imtr4g 261 | . . 3 ⊢ (A ⊆ B → (yAx → yBx)) |
5 | 4 | ssopab2dv 4715 | . 2 ⊢ (A ⊆ B → {〈x, y〉 ∣ yAx} ⊆ {〈x, y〉 ∣ yBx}) |
6 | df-cnv 4785 | . 2 ⊢ ◡A = {〈x, y〉 ∣ yAx} | |
7 | df-cnv 4785 | . 2 ⊢ ◡B = {〈x, y〉 ∣ yBx} | |
8 | 5, 6, 7 | 3sstr4g 3312 | 1 ⊢ (A ⊆ B → ◡A ⊆ ◡B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ⊆ wss 3257 〈cop 4561 {copab 4622 class class class wbr 4639 ◡ccnv 4771 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-ss 3259 df-opab 4623 df-br 4640 df-cnv 4785 |
This theorem is referenced by: cnveq 4886 rnss 4959 cnvtr 5098 funss 5126 funcnvuni 5161 funres11 5164 funcnvres 5165 foimacnv 5303 |
Copyright terms: Public domain | W3C validator |