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Theorem mpst123 31437
Description: Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mpstssv.p |- P = (mPreSt ` T )
Assertion
Ref Expression
mpst123 |- ( X e. P -> X = <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) , ( 2nd ` X ) >. )
Proof of Theorem mpst123
StepHypRef Expression
1 mpstssv.p . . . 4 |- P = (mPreSt ` T )
21mpstssv 31436 . . 3 |- P C_ ( ( _V X. _V ) X. _V )
32sseli 3599 . 2 |- ( X e. P -> X e. ( ( _V X. _V ) X. _V ) )
4 1st2nd2 7205 . . . 4 |- ( X e. ( ( _V X. _V ) X. _V ) -> X = <. ( 1st ` X ) , ( 2nd ` X ) >. )
5 xp1st 7198 . . . . . 6 |- ( X e. ( ( _V X. _V ) X. _V ) -> ( 1st ` X ) e. ( _V X. _V ) )
6 1st2nd2 7205 . . . . . 6 |- ( ( 1st ` X ) e. ( _V X. _V ) -> ( 1st ` X ) = <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) >. )
75, 6syl 17 . . . . 5 |- ( X e. ( ( _V X. _V ) X. _V ) -> ( 1st ` X ) = <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) >. )
87opeq1d 4408 . . . 4 |- ( X e. ( ( _V X. _V ) X. _V ) -> <. ( 1st ` X ) , ( 2nd ` X ) >. = <. <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) >. , ( 2nd ` X ) >. )
94, 8eqtrd 2656 . . 3 |- ( X e. ( ( _V X. _V ) X. _V ) -> X = <. <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) >. , ( 2nd ` X ) >. )
10 df-ot 4186 . . 3 |- <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) , ( 2nd ` X ) >. = <. <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) >. , ( 2nd ` X ) >.
119, 10syl6eqr 2674 . 2 |- ( X e. ( ( _V X. _V ) X. _V ) -> X = <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) , ( 2nd ` X ) >. )
123, 11syl 17 1 |- ( X e. P -> X = <. ( 1st ` ( 1st ` X ) ) , ( 2nd ` ( 1st ` X ) ) , ( 2nd ` X ) >. )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   <.cotp 4185   X. cxp 5112   ` cfv 5888   1stc1st 7166   2ndc2nd 7167  mPreStcmpst 31370
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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169  df-mpst 31390
This theorem is referenced by:  msrf  31439  msrid  31442  mthmpps  31479
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