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Theorem cvmlift2lem4 31288
Description: Lemma for cvmlift2 31298. (Contributed by Mario Carneiro, 1-Jun-2015.)
Hypotheses
Ref Expression
cvmlift2.b |- B = U. C
cvmlift2.f |- ( ph -> F e. ( C CovMap J ) )
cvmlift2.g |- ( ph -> G e. ( ( II tX II ) Cn J ) )
cvmlift2.p |- ( ph -> P e. B )
cvmlift2.i |- ( ph -> ( F ` P ) = ( 0 G 0 ) )
cvmlift2.h |- H = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( z G 0 ) ) /\ ( f ` 0 ) = P ) )
cvmlift2.k |- K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) )
Assertion
Ref Expression
cvmlift2lem4 |- ( ( X e. ( 0 [,] 1 ) /\ Y e. ( 0 [,] 1 ) ) -> ( X K Y ) = ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` Y ) )
Distinct variable groups:   x, f, y, z, F   ph, f, x, y, z   f, J, x, y, z   f, G, x, y, z   f, H, x, y, z   f, X, x, y, z   C, f, x, y, z   P, f, x, y, z   x, B, y, z   f, Y, x, y, z   f, K, x, y, z
Allowed substitution hint:   B( f)
Proof of Theorem cvmlift2lem4
StepHypRef Expression
1 oveq1 6657 . . . . . . 7 |- ( x = X -> ( x G z ) = ( X G z ) )
21mpteq2dv 4745 . . . . . 6 |- ( x = X -> ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) )
32eqeq2d 2632 . . . . 5 |- ( x = X -> ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) <-> ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) ) )
4 fveq2 6191 . . . . . 6 |- ( x = X -> ( H ` x ) = ( H ` X ) )
54eqeq2d 2632 . . . . 5 |- ( x = X -> ( ( f ` 0 ) = ( H ` x ) <-> ( f ` 0 ) = ( H ` X ) ) )
63, 5anbi12d 747 . . . 4 |- ( x = X -> ( ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) <-> ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) )
76riotabidv 6613 . . 3 |- ( x = X -> ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) = ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) )
87fveq1d 6193 . 2 |- ( x = X -> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) = ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` y ) )
9 fveq2 6191 . 2 |- ( y = Y -> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` y ) = ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` Y ) )
10 cvmlift2.k . 2 |- K = ( x e. ( 0 [,] 1 ) , y e. ( 0 [,] 1 ) |-> ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( x G z ) ) /\ ( f ` 0 ) = ( H ` x ) ) ) ` y ) )
11 fvex 6201 . 2 |- ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` Y ) e. _V
128, 9, 10, 11ovmpt2 6796 1 |- ( ( X e. ( 0 [,] 1 ) /\ Y e. ( 0 [,] 1 ) ) -> ( X K Y ) = ( ( iota_ f e. ( II Cn C ) ( ( F o. f ) = ( z e. ( 0 [,] 1 ) |-> ( X G z ) ) /\ ( f ` 0 ) = ( H ` X ) ) ) ` Y ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   U.cuni 4436   |-> cmpt 4729   o. ccom 5118   ` cfv 5888   iota_crio 6610  (class class class)co 6650   |-> cmpt2 6652   0cc0 9936   1c1 9937   [,]cicc 12178   Cn ccn 21028   tX ctx 21363   IIcii 22678   CovMap ccvm 31237
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  ax-sep 4781  ax-nul 4789  ax-pr 4906
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-sn 4178  df-pr 4180  df-op 4184  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-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655
This theorem is referenced by:  cvmlift2lem6  31290  cvmlift2lem8  31292
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