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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 5851 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶) |
3 | reseqi.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | reseq2i 5852 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
5 | 2, 4 | eqtri 2846 | 1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ↾ cres 5559 |
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-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-rab 3149 df-in 3945 df-opab 5131 df-xp 5563 df-res 5569 |
This theorem is referenced by: cnvresid 6435 wfrlem5 7961 dfoi 8977 lubfval 17590 glbfval 17603 oduglb 17751 odulub 17753 dvlog 25236 dvlog2 25238 issubgr 27055 finsumvtxdg2size 27334 sitgclg 31602 fprlem1 33139 frrlem15 33144 fourierdlem57 42455 fourierdlem74 42472 fourierdlem75 42473 |
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