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Mirrors > Home > NFE Home > Th. List > cnveq | GIF version |
Description: Equality theorem for converse. (Contributed by set.mm contributors, 13-Aug-1995.) |
Ref | Expression |
---|---|
cnveq | ⊢ (A = B → ◡A = ◡B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvss 4885 | . . 3 ⊢ (A ⊆ B → ◡A ⊆ ◡B) | |
2 | cnvss 4885 | . . 3 ⊢ (B ⊆ A → ◡B ⊆ ◡A) | |
3 | 1, 2 | anim12i 549 | . 2 ⊢ ((A ⊆ B ∧ B ⊆ A) → (◡A ⊆ ◡B ∧ ◡B ⊆ ◡A)) |
4 | eqss 3287 | . 2 ⊢ (A = B ↔ (A ⊆ B ∧ B ⊆ A)) | |
5 | eqss 3287 | . 2 ⊢ (◡A = ◡B ↔ (◡A ⊆ ◡B ∧ ◡B ⊆ ◡A)) | |
6 | 3, 4, 5 | 3imtr4i 257 | 1 ⊢ (A = B → ◡A = ◡B) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ⊆ wss 3257 ◡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: cnveqi 4887 cnveqd 4888 cnveqb 5063 funcnvuni 5161 f1eq1 5253 f1o00 5317 enprmaplem3 6078 enprmaplem5 6080 enprmaplem6 6081 |
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