Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvhvscacbv Structured version   Visualization version   GIF version

Theorem dvhvscacbv 40061
Description: Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)
Hypothesis
Ref Expression
dvhvscaval.s ยท = (๐‘  โˆˆ ๐ธ, ๐‘“ โˆˆ (๐‘‡ ร— ๐ธ) โ†ฆ โŸจ(๐‘ โ€˜(1st โ€˜๐‘“)), (๐‘  โˆ˜ (2nd โ€˜๐‘“))โŸฉ)
Assertion
Ref Expression
dvhvscacbv ยท = (๐‘ก โˆˆ ๐ธ, ๐‘” โˆˆ (๐‘‡ ร— ๐ธ) โ†ฆ โŸจ(๐‘กโ€˜(1st โ€˜๐‘”)), (๐‘ก โˆ˜ (2nd โ€˜๐‘”))โŸฉ)
Distinct variable groups:   ๐‘“,๐‘ ,๐‘ก,๐‘”,๐ธ   ๐‘‡,๐‘ ,๐‘“,๐‘ก,๐‘”
Allowed substitution hints:   ยท (๐‘ก,๐‘“,๐‘”,๐‘ )

Proof of Theorem dvhvscacbv
StepHypRef Expression
1 dvhvscaval.s . 2 ยท = (๐‘  โˆˆ ๐ธ, ๐‘“ โˆˆ (๐‘‡ ร— ๐ธ) โ†ฆ โŸจ(๐‘ โ€˜(1st โ€˜๐‘“)), (๐‘  โˆ˜ (2nd โ€˜๐‘“))โŸฉ)
2 fveq1 6890 . . . 4 (๐‘  = ๐‘ก โ†’ (๐‘ โ€˜(1st โ€˜๐‘“)) = (๐‘กโ€˜(1st โ€˜๐‘“)))
3 coeq1 5857 . . . 4 (๐‘  = ๐‘ก โ†’ (๐‘  โˆ˜ (2nd โ€˜๐‘“)) = (๐‘ก โˆ˜ (2nd โ€˜๐‘“)))
42, 3opeq12d 4881 . . 3 (๐‘  = ๐‘ก โ†’ โŸจ(๐‘ โ€˜(1st โ€˜๐‘“)), (๐‘  โˆ˜ (2nd โ€˜๐‘“))โŸฉ = โŸจ(๐‘กโ€˜(1st โ€˜๐‘“)), (๐‘ก โˆ˜ (2nd โ€˜๐‘“))โŸฉ)
5 2fveq3 6896 . . . 4 (๐‘“ = ๐‘” โ†’ (๐‘กโ€˜(1st โ€˜๐‘“)) = (๐‘กโ€˜(1st โ€˜๐‘”)))
6 fveq2 6891 . . . . 5 (๐‘“ = ๐‘” โ†’ (2nd โ€˜๐‘“) = (2nd โ€˜๐‘”))
76coeq2d 5862 . . . 4 (๐‘“ = ๐‘” โ†’ (๐‘ก โˆ˜ (2nd โ€˜๐‘“)) = (๐‘ก โˆ˜ (2nd โ€˜๐‘”)))
85, 7opeq12d 4881 . . 3 (๐‘“ = ๐‘” โ†’ โŸจ(๐‘กโ€˜(1st โ€˜๐‘“)), (๐‘ก โˆ˜ (2nd โ€˜๐‘“))โŸฉ = โŸจ(๐‘กโ€˜(1st โ€˜๐‘”)), (๐‘ก โˆ˜ (2nd โ€˜๐‘”))โŸฉ)
94, 8cbvmpov 7506 . 2 (๐‘  โˆˆ ๐ธ, ๐‘“ โˆˆ (๐‘‡ ร— ๐ธ) โ†ฆ โŸจ(๐‘ โ€˜(1st โ€˜๐‘“)), (๐‘  โˆ˜ (2nd โ€˜๐‘“))โŸฉ) = (๐‘ก โˆˆ ๐ธ, ๐‘” โˆˆ (๐‘‡ ร— ๐ธ) โ†ฆ โŸจ(๐‘กโ€˜(1st โ€˜๐‘”)), (๐‘ก โˆ˜ (2nd โ€˜๐‘”))โŸฉ)
101, 9eqtri 2760 1 ยท = (๐‘ก โˆˆ ๐ธ, ๐‘” โˆˆ (๐‘‡ ร— ๐ธ) โ†ฆ โŸจ(๐‘กโ€˜(1st โ€˜๐‘”)), (๐‘ก โˆ˜ (2nd โ€˜๐‘”))โŸฉ)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  โŸจcop 4634   ร— cxp 5674   โˆ˜ ccom 5680  โ€˜cfv 6543   โˆˆ cmpo 7413  1st c1st 7975  2nd c2nd 7976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-co 5685  df-iota 6495  df-fv 6551  df-oprab 7415  df-mpo 7416
This theorem is referenced by:  dvhvscaval  40062
  Copyright terms: Public domain W3C validator