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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 5996 | . 2 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐶) |
3 | reseqi.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | reseq2i 5997 | . 2 ⊢ (𝐵 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
5 | 2, 4 | eqtri 2763 | 1 ⊢ (𝐴 ↾ 𝐶) = (𝐵 ↾ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ↾ cres 5691 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-rab 3434 df-in 3970 df-opab 5211 df-xp 5695 df-res 5701 |
This theorem is referenced by: cnvresid 6647 fprlem1 8324 wfrlem5OLD 8352 dfoi 9549 frrlem15 9795 lubfval 18408 glbfval 18421 odulub 18465 oduglb 18467 dvlog 26708 dvlog2 26710 issubgr 29303 finsumvtxdg2size 29583 sitgclg 34324 fourierdlem57 46119 fourierdlem74 46136 fourierdlem75 46137 |
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