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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 4023 | . . 3 ⊢ (𝐹 = 𝐺 → 𝐹 ⊆ 𝐺) | |
2 | tposss 7893 | . . 3 ⊢ (𝐹 ⊆ 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐹 = 𝐺 → tpos 𝐹 ⊆ tpos 𝐺) |
4 | eqimss2 4024 | . . 3 ⊢ (𝐹 = 𝐺 → 𝐺 ⊆ 𝐹) | |
5 | tposss 7893 | . . 3 ⊢ (𝐺 ⊆ 𝐹 → tpos 𝐺 ⊆ tpos 𝐹) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐹 = 𝐺 → tpos 𝐺 ⊆ tpos 𝐹) |
7 | 3, 6 | eqssd 3984 | 1 ⊢ (𝐹 = 𝐺 → tpos 𝐹 = tpos 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ⊆ wss 3936 tpos ctpos 7891 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-res 5567 df-tpos 7892 |
This theorem is referenced by: tposeqd 7895 tposeqi 7925 |
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