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Theorem dvhvscacbv 39564
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 6842 . . . 4 (๐‘  = ๐‘ก โ†’ (๐‘ โ€˜(1st โ€˜๐‘“)) = (๐‘กโ€˜(1st โ€˜๐‘“)))
3 coeq1 5814 . . . 4 (๐‘  = ๐‘ก โ†’ (๐‘  โˆ˜ (2nd โ€˜๐‘“)) = (๐‘ก โˆ˜ (2nd โ€˜๐‘“)))
42, 3opeq12d 4839 . . 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 6848 . . . 4 (๐‘“ = ๐‘” โ†’ (๐‘กโ€˜(1st โ€˜๐‘“)) = (๐‘กโ€˜(1st โ€˜๐‘”)))
6 fveq2 6843 . . . . 5 (๐‘“ = ๐‘” โ†’ (2nd โ€˜๐‘“) = (2nd โ€˜๐‘”))
76coeq2d 5819 . . . 4 (๐‘“ = ๐‘” โ†’ (๐‘ก โˆ˜ (2nd โ€˜๐‘“)) = (๐‘ก โˆ˜ (2nd โ€˜๐‘”)))
85, 7opeq12d 4839 . . 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 7453 . 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 2765 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 1542  โŸจcop 4593   ร— cxp 5632   โˆ˜ ccom 5638  โ€˜cfv 6497   โˆˆ cmpo 7360  1st c1st 7920  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-co 5643  df-iota 6449  df-fv 6505  df-oprab 7362  df-mpo 7363
This theorem is referenced by:  dvhvscaval  39565
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