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| Mirrors > Home > MPE Home > Th. List > reseq12i | Structured version Visualization version GIF version | ||
| Description: Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.) |
| Ref | Expression |
|---|---|
| reseqi.1 | ⊢ 𝐴 = 𝐵 |
| reseqi.2 | ⊢ 𝐶 = 𝐷 |
| Ref | Expression |
|---|---|
| reseq12i | ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | reseqi.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
| 2 | 1 | reseq1i 5993 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶) |
| 3 | reseqi.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
| 4 | 3 | reseq2i 5994 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
| 5 | 2, 4 | eqtri 2765 | 1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ↾ cres 5687 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-in 3958 df-opab 5206 df-xp 5691 df-res 5697 |
| This theorem is referenced by: cnvresid 6645 fprlem1 8325 wfrlem5OLD 8353 dfoi 9551 frrlem15 9797 lubfval 18395 glbfval 18408 odulub 18452 oduglb 18454 dvlog 26693 dvlog2 26695 issubgr 29288 finsumvtxdg2size 29568 sitgclg 34344 fourierdlem57 46178 fourierdlem74 46195 fourierdlem75 46196 |
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