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Mirrors > Home > MPE Home > Th. List > tposeq | Structured version Visualization version 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 3943 | . . 3 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺) | |
2 | tposss 7947 | . . 3 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) |
4 | eqimss2 3944 | . . 3 ⊢ (𝐹 = 𝐺 → 𝐺 ⊆ 𝐹) | |
5 | tposss 7947 | . . 3 ⊢ (𝐺 ⊆ 𝐹 → tpos 𝐺 ⊆ tpos 𝐹) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹) |
7 | 3, 6 | eqssd 3904 | 1 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ⊆ wss 3853 tpos ctpos 7945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-mpt 5121 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-res 5548 df-tpos 7946 |
This theorem is referenced by: tposeqd 7949 tposeqi 7979 |
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