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Theorem hdmap1cbv 37092
Description: Frequently used lemma to change bound variables in L hypothesis. (Contributed by NM, 15-May-2015.)
Hypothesis
Ref Expression
hdmap1cbv.l |- L = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
Assertion
Ref Expression
hdmap1cbv |- L = ( y e. _V |-> if ( ( 2nd ` y ) = .0. , Q , ( iota_ i e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) ) ) )
Distinct variable groups:   h, i, x, y, D   h, J, i, x, y   h, M, i, x, y   h, N, i, x, y   x, .0. , y   x, Q, y   R, h, i, x, y   .- , h, i, x, y
Allowed substitution hints:   Q( h, i)   L( x, y, h, i)   .0. ( h, i)
Proof of Theorem hdmap1cbv
StepHypRef Expression
1 hdmap1cbv.l . 2 |- L = ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) )
2 fveq2 6191 . . . . 5 |- ( x = y -> ( 2nd ` x ) = ( 2nd ` y ) )
32eqeq1d 2624 . . . 4 |- ( x = y -> ( ( 2nd ` x ) = .0. <-> ( 2nd ` y ) = .0. ) )
42sneqd 4189 . . . . . . . . 9 |- ( x = y -> { ( 2nd ` x ) } = { ( 2nd ` y ) } )
54fveq2d 6195 . . . . . . . 8 |- ( x = y -> ( N ` { ( 2nd ` x ) } ) = ( N ` { ( 2nd ` y ) } ) )
65fveq2d 6195 . . . . . . 7 |- ( x = y -> ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( M ` ( N ` { ( 2nd ` y ) } ) ) )
76eqeq1d 2624 . . . . . 6 |- ( x = y -> ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) ) )
8 fveq2 6191 . . . . . . . . . . . 12 |- ( x = y -> ( 1st ` x ) = ( 1st ` y ) )
98fveq2d 6195 . . . . . . . . . . 11 |- ( x = y -> ( 1st ` ( 1st ` x ) ) = ( 1st ` ( 1st ` y ) ) )
109, 2oveq12d 6668 . . . . . . . . . 10 |- ( x = y -> ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) = ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) )
1110sneqd 4189 . . . . . . . . 9 |- ( x = y -> { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } = { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } )
1211fveq2d 6195 . . . . . . . 8 |- ( x = y -> ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) = ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) )
1312fveq2d 6195 . . . . . . 7 |- ( x = y -> ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) )
148fveq2d 6195 . . . . . . . . . 10 |- ( x = y -> ( 2nd ` ( 1st ` x ) ) = ( 2nd ` ( 1st ` y ) ) )
1514oveq1d 6665 . . . . . . . . 9 |- ( x = y -> ( ( 2nd ` ( 1st ` x ) ) R h ) = ( ( 2nd ` ( 1st ` y ) ) R h ) )
1615sneqd 4189 . . . . . . . 8 |- ( x = y -> { ( ( 2nd ` ( 1st ` x ) ) R h ) } = { ( ( 2nd ` ( 1st ` y ) ) R h ) } )
1716fveq2d 6195 . . . . . . 7 |- ( x = y -> ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) )
1813, 17eqeq12d 2637 . . . . . 6 |- ( x = y -> ( ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) <-> ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) )
197, 18anbi12d 747 . . . . 5 |- ( x = y -> ( ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) <-> ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) )
2019riotabidv 6613 . . . 4 |- ( x = y -> ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) = ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) )
213, 20ifbieq2d 4111 . . 3 |- ( x = y -> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) = if ( ( 2nd ` y ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) ) )
2221cbvmptv 4750 . 2 |- ( x e. _V |-> if ( ( 2nd ` x ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` x ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` x ) ) .- ( 2nd ` x ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` x ) ) R h ) } ) ) ) ) ) = ( y e. _V |-> if ( ( 2nd ` y ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) ) )
23 sneq 4187 . . . . . . . 8 |- ( h = i -> { h } = { i } )
2423fveq2d 6195 . . . . . . 7 |- ( h = i -> ( J ` { h } ) = ( J ` { i } ) )
2524eqeq2d 2632 . . . . . 6 |- ( h = i -> ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) <-> ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) ) )
26 oveq2 6658 . . . . . . . . 9 |- ( h = i -> ( ( 2nd ` ( 1st ` y ) ) R h ) = ( ( 2nd ` ( 1st ` y ) ) R i ) )
2726sneqd 4189 . . . . . . . 8 |- ( h = i -> { ( ( 2nd ` ( 1st ` y ) ) R h ) } = { ( ( 2nd ` ( 1st ` y ) ) R i ) } )
2827fveq2d 6195 . . . . . . 7 |- ( h = i -> ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) )
2928eqeq2d 2632 . . . . . 6 |- ( h = i -> ( ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) <-> ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) )
3025, 29anbi12d 747 . . . . 5 |- ( h = i -> ( ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) <-> ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) ) )
3130cbvriotav 6622 . . . 4 |- ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) = ( iota_ i e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) )
32 ifeq2 4091 . . . 4 |- ( ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) = ( iota_ i e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) ) -> if ( ( 2nd ` y ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) ) = if ( ( 2nd ` y ) = .0. , Q , ( iota_ i e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) ) ) )
3331, 32ax-mp 5 . . 3 |- if ( ( 2nd ` y ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) ) = if ( ( 2nd ` y ) = .0. , Q , ( iota_ i e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) ) )
3433mpteq2i 4741 . 2 |- ( y e. _V |-> if ( ( 2nd ` y ) = .0. , Q , ( iota_ h e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { h } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R h ) } ) ) ) ) ) = ( y e. _V |-> if ( ( 2nd ` y ) = .0. , Q , ( iota_ i e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) ) ) )
351, 22, 343eqtri 2648 1 |- L = ( y e. _V |-> if ( ( 2nd ` y ) = .0. , Q , ( iota_ i e. D ( ( M ` ( N ` { ( 2nd ` y ) } ) ) = ( J ` { i } ) /\ ( M ` ( N ` { ( ( 1st ` ( 1st ` y ) ) .- ( 2nd ` y ) ) } ) ) = ( J ` { ( ( 2nd ` ( 1st ` y ) ) R i ) } ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   /\ wa 384   = wceq 1483   _Vcvv 3200   ifcif 4086   {csn 4177   |-> cmpt 4729   ` cfv 5888   iota_crio 6610  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167
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
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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-iota 5851  df-fv 5896  df-riota 6611  df-ov 6653
This theorem is referenced by:  hdmap1valc  37093  hdmap1eu  37115  hdmap1euOLDN  37116
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