Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5392 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶) |
3 | reseqi.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | reseq2i 5393 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
5 | 2, 4 | eqtri 2644 | 1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ↾ cres 5116 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-in 3581 df-opab 4713 df-xp 5120 df-res 5126 |
This theorem is referenced by: cnvresid 5968 wfrlem5 7419 dfoi 8416 lubfval 16978 glbfval 16991 oduglb 17139 odulub 17141 dvlog 24397 dvlog2 24399 issubgr 26163 finsumvtxdg2size 26446 sitgclg 30404 frrlem5 31784 fourierdlem57 40380 fourierdlem74 40397 fourierdlem75 40398 |
Copyright terms: Public domain | W3C validator |