Theorem fnwe 7293

Description: A variant on lexicographic order, which sorts first by some function of the base set, and then by a "backup" well-ordering when the function value is equal on both elements. (Contributed by Mario Carneiro, 10-Mar-2013.) (Revised by Mario Carneiro, 18-Nov-2014.)

Hypotheses

Ref	Expression
fnwe.1	|- T = { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) }
fnwe.2	|- ( ph -> F : A --> B )
fnwe.3	|- ( ph -> R We B )
fnwe.4	|- ( ph -> S We A )
fnwe.5	|- ( ph -> ( F " w ) e. _V )

Assertion

Ref	Expression
fnwe	|- ( ph -> T We A )

Distinct variable groups:   x, w, y, A   w, B, x, y   ph, w, x   w, F, x, y   w, R, x, y   w, S, x, y   w, T

Allowed substitution hints:   ph( y)   T( x, y)

Proof of Theorem fnwe

Dummy variables v u z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnwe.1	. 2 |- T = { <. x , y >. | ( ( x e. A /\ y e. A ) /\ ( ( F ` x ) R ( F ` y ) \/ ( ( F ` x ) = ( F ` y ) /\ x S y ) ) ) }
2		fnwe.2	. 2 |- ( ph -> F : A --> B )
3		fnwe.3	. 2 |- ( ph -> R We B )
4		fnwe.4	. 2 |- ( ph -> S We A )
5		fnwe.5	. 2 |- ( ph -> ( F " w ) e. _V )
6		eqid 2622	. 2 |- { <. u , v >. | ( ( u e. ( B X. A ) /\ v e. ( B X. A ) ) /\ ( ( 1st ` u ) R ( 1st ` v ) \/ ( ( 1st ` u ) = ( 1st ` v ) /\ ( 2nd ` u ) S ( 2nd ` v ) ) ) ) } = { <. u , v >. | ( ( u e. ( B X. A ) /\ v e. ( B X. A ) ) /\ ( ( 1st ` u ) R ( 1st ` v ) \/ ( ( 1st ` u ) = ( 1st ` v ) /\ ( 2nd ` u ) S ( 2nd ` v ) ) ) ) }
7		eqid 2622	. 2 |- ( z e. A |-> <. ( F ` z ) , z >. ) = ( z e. A |-> <. ( F ` z ) , z >. )
8	1, 2, 3, 4, 5, 6, 7	fnwelem 7292	1 |- ( ph -> T We A )

Syntax hints:   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   class class class wbr 4653   {copab 4712   |-> cmpt 4729   We wwe 5072   X. cxp 5112   "cima 5117   -->wf 5884   ` cfv 5888   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-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-3or 1038  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-ne 2795  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-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-1st 7168  df-2nd 7169

This theorem is referenced by:  r0weon  8835