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Theorem noseponlem 30626
Description: Lemma for nosepon 30627. Consider a case of proper subset domain. (Contributed by Scott Fenton, 21-Sep-2020.)
Assertion
Ref Expression
noseponlem  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  -.  A. x  e.  On  ( A `  x )  =  ( B `  x ) )
Distinct variable groups:    x, A    x, B

Proof of Theorem noseponlem
StepHypRef Expression
1 nodmon 30608 . . . 4  |-  ( A  e.  No  ->  dom  A  e.  On )
213ad2ant1 1051 . . 3  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  dom  A  e.  On )
3 nodmord 30611 . . . . . . 7  |-  ( A  e.  No  ->  Ord  dom 
A )
4 ordirr 5448 . . . . . . 7  |-  ( Ord 
dom  A  ->  -.  dom  A  e.  dom  A )
53, 4syl 17 . . . . . 6  |-  ( A  e.  No  ->  -.  dom  A  e.  dom  A
)
653ad2ant1 1051 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  -.  dom  A  e. 
dom  A )
7 ndmfv 5903 . . . . 5  |-  ( -. 
dom  A  e.  dom  A  ->  ( A `  dom  A )  =  (/) )
86, 7syl 17 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  ( A `  dom  A )  =  (/) )
9 nosgnn0 30616 . . . . . . 7  |-  -.  (/)  e.  { 1o ,  2o }
10 elno3 30613 . . . . . . . . . . 11  |-  ( B  e.  No  <->  ( B : dom  B --> { 1o ,  2o }  /\  dom  B  e.  On ) )
1110simplbi 467 . . . . . . . . . 10  |-  ( B  e.  No  ->  B : dom  B --> { 1o ,  2o } )
12113ad2ant2 1052 . . . . . . . . 9  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  B : dom  B --> { 1o ,  2o } )
13 simp3 1032 . . . . . . . . 9  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  dom  A  e.  dom  B )
1412, 13ffvelrnd 6038 . . . . . . . 8  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  ( B `  dom  A )  e.  { 1o ,  2o } )
15 eleq1 2537 . . . . . . . 8  |-  ( ( B `  dom  A
)  =  (/)  ->  (
( B `  dom  A )  e.  { 1o ,  2o }  <->  (/)  e.  { 1o ,  2o } ) )
1614, 15syl5ibcom 228 . . . . . . 7  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  ( ( B `
 dom  A )  =  (/)  ->  (/)  e.  { 1o ,  2o } ) )
179, 16mtoi 183 . . . . . 6  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  -.  ( B `  dom  A )  =  (/) )
1817neqned 2650 . . . . 5  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  ( B `  dom  A )  =/=  (/) )
1918necomd 2698 . . . 4  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  (/)  =/=  ( B `
 dom  A )
)
208, 19eqnetrd 2710 . . 3  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  ( A `  dom  A )  =/=  ( B `  dom  A ) )
21 fveq2 5879 . . . . 5  |-  ( x  =  dom  A  -> 
( A `  x
)  =  ( A `
 dom  A )
)
22 fveq2 5879 . . . . 5  |-  ( x  =  dom  A  -> 
( B `  x
)  =  ( B `
 dom  A )
)
2321, 22neeq12d 2704 . . . 4  |-  ( x  =  dom  A  -> 
( ( A `  x )  =/=  ( B `  x )  <->  ( A `  dom  A
)  =/=  ( B `
 dom  A )
) )
2423rspcev 3136 . . 3  |-  ( ( dom  A  e.  On  /\  ( A `  dom  A )  =/=  ( B `
 dom  A )
)  ->  E. x  e.  On  ( A `  x )  =/=  ( B `  x )
)
252, 20, 24syl2anc 673 . 2  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  E. x  e.  On  ( A `  x )  =/=  ( B `  x ) )
26 df-ne 2643 . . . 4  |-  ( ( A `  x )  =/=  ( B `  x )  <->  -.  ( A `  x )  =  ( B `  x ) )
2726rexbii 2881 . . 3  |-  ( E. x  e.  On  ( A `  x )  =/=  ( B `  x
)  <->  E. x  e.  On  -.  ( A `  x
)  =  ( B `
 x ) )
28 rexnal 2836 . . 3  |-  ( E. x  e.  On  -.  ( A `  x )  =  ( B `  x )  <->  -.  A. x  e.  On  ( A `  x )  =  ( B `  x ) )
2927, 28bitri 257 . 2  |-  ( E. x  e.  On  ( A `  x )  =/=  ( B `  x
)  <->  -.  A. x  e.  On  ( A `  x )  =  ( B `  x ) )
3025, 29sylib 201 1  |-  ( ( A  e.  No  /\  B  e.  No  /\  dom  A  e.  dom  B )  ->  -.  A. x  e.  On  ( A `  x )  =  ( B `  x ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   E.wrex 2757   (/)c0 3722   {cpr 3961   dom cdm 4839   Ord word 5429   Oncon0 5430   -->wf 5585   ` cfv 5589   1oc1o 7193   2oc2o 7194   Nocsur 30598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-ord 5433  df-on 5434  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-1o 7200  df-2o 7201  df-no 30601
This theorem is referenced by:  nosepon  30627
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