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Mirrors > Home > ILE Home > Th. List > tposeq | GIF version |
Description: Equality theorem for transposition. (Contributed by Mario Carneiro, 10-Sep-2015.) |
Ref | Expression |
---|---|
tposeq | ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss 3101 | . . 3 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺) | |
2 | tposss 6073 | . . 3 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) |
4 | eqimss2 3102 | . . 3 ⊢ (𝐹 = 𝐺 → 𝐺 ⊆ 𝐹) | |
5 | tposss 6073 | . . 3 ⊢ (𝐺 ⊆ 𝐹 → tpos 𝐺 ⊆ tpos 𝐹) | |
6 | 4, 5 | syl 14 | . 2 ⊢ (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹) |
7 | 3, 6 | eqssd 3064 | 1 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1299 ⊆ wss 3021 tpos ctpos 6071 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-br 3876 df-opab 3930 df-mpt 3931 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-res 4489 df-tpos 6072 |
This theorem is referenced by: tposeqd 6075 tposeqi 6104 |
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