Step | Hyp | Ref
| Expression |
1 | | 1stfcl.t |
. . . 4
 c   |
2 | | eqid 2451 |
. . . . 5
         |
3 | | eqid 2451 |
. . . . 5
         |
4 | 1, 2, 3 | xpcbas 16063 |
. . . 4
               |
5 | | eqid 2451 |
. . . 4
       |
6 | | 1stfcl.c |
. . . 4
   |
7 | | 1stfcl.d |
. . . 4
   |
8 | | 1stfcl.p |
. . . 4
 F
  |
9 | 1, 4, 5, 6, 7, 8 | 1stfval 16076 |
. . 3
                                     
  
         |
10 | | fo1st 6813 |
. . . . . . . 8
     |
11 | | fofun 5794 |
. . . . . . . 8
       |
12 | 10, 11 | ax-mp 5 |
. . . . . . 7
 |
13 | | fvex 5875 |
. . . . . . . 8
     |
14 | | fvex 5875 |
. . . . . . . 8
     |
15 | 13, 14 | xpex 6595 |
. . . . . . 7
           |
16 | | resfunexg 6130 |
. . . . . . 7
 
                         |
17 | 12, 15, 16 | mp2an 678 |
. . . . . 6
             |
18 | 15, 15 | mpt2ex 6870 |
. . . . . 6
                      
  
       |
19 | 17, 18 | op2ndd 6804 |
. . . . 5
                                                                                      |
20 | 9, 19 | syl 17 |
. . . 4
                           
  
        |
21 | 20 | opeq2d 4173 |
. . 3
                                                        
  
         |
22 | 9, 21 | eqtr4d 2488 |
. 2
                      |
23 | | eqid 2451 |
. . . 4
       |
24 | | eqid 2451 |
. . . 4
         |
25 | | eqid 2451 |
. . . 4
         |
26 | | eqid 2451 |
. . . 4
comp  comp   |
27 | | eqid 2451 |
. . . 4
comp  comp   |
28 | 1, 6, 7 | xpccat 16075 |
. . . 4
   |
29 | | f1stres 6815 |
. . . . 5
                               |
30 | 29 | a1i 11 |
. . . 4
                                 |
31 | | eqid 2451 |
. . . . . 6
                      
  
                                       |
32 | | ovex 6318 |
. . . . . . 7
        |
33 | | resfunexg 6130 |
. . . . . . 7
 
                   |
34 | 12, 32, 33 | mp2an 678 |
. . . . . 6
          |
35 | 31, 34 | fnmpt2i 6862 |
. . . . 5
                      
  
                             |
36 | 20 | fneq1d 5666 |
. . . . 5
                                                                                     |
37 | 35, 36 | mpbiri 237 |
. . . 4
                             |
38 | | f1stres 6815 |
. . . . . 6
                                                                                      |
39 | 6 | adantr 467 |
. . . . . . . . 9
 
          
              |
40 | 7 | adantr 467 |
. . . . . . . . 9
 
          
              |
41 | | simprl 764 |
. . . . . . . . 9
 
          
                        |
42 | | simprr 766 |
. . . . . . . . 9
 
          
                        |
43 | 1, 4, 5, 39, 40, 8, 41, 42 | 1stf2 16078 |
. . . . . . . 8
 
          
                    
  
       |
44 | | eqid 2451 |
. . . . . . . . . 10
       |
45 | 1, 4, 23, 44, 5, 41, 42 | xpchom 16065 |
. . . . . . . . 9
 
          
                                                     |
46 | 45 | reseq2d 5105 |
. . . . . . . 8
 
          
                                                         |
47 | 43, 46 | eqtrd 2485 |
. . . . . . 7
 
          
                    
                                   |
48 | 47 | feq1d 5714 |
. . . . . 6
 
          
                                                                       
                                                                                        |
49 | 38, 48 | mpbiri 237 |
. . . . 5
 
          
                                                                         |
50 | | fvres 5879 |
. . . . . . . 8
                                 |
51 | 50 | ad2antrl 734 |
. . . . . . 7
 
          
                                  |
52 | | fvres 5879 |
. . . . . . . 8
                                 |
53 | 52 | ad2antll 735 |
. . . . . . 7
 
          
                                  |
54 | 51, 53 | oveq12d 6308 |
. . . . . 6
 
          
              
                    
                                |
55 | 45, 54 | feq23d 5723 |
. . . . 5
 
          
                                                                      
                                                              |
56 | 49, 55 | mpbird 236 |
. . . 4
 
          
                                                      
                 |
57 | 28 | adantr 467 |
. . . . . . . 8
 
             |
58 | | simpr 463 |
. . . . . . . 8
 
                       |
59 | 4, 5, 24, 57, 58 | catidcl 15588 |
. . . . . . 7
 
                            |
60 | | fvres 5879 |
. . . . . . 7
                                                   |
61 | 59, 60 | syl 17 |
. . . . . 6
 
                                              |
62 | | 1st2nd2 6830 |
. . . . . . . . . 10
                        |
63 | 62 | adantl 468 |
. . . . . . . . 9
 
                        |
64 | 63 | fveq2d 5869 |
. . . . . . . 8
 
                                        |
65 | 6 | adantr 467 |
. . . . . . . . 9
 
             |
66 | 7 | adantr 467 |
. . . . . . . . 9
 
             |
67 | | eqid 2451 |
. . . . . . . . 9
         |
68 | | xp1st 6823 |
. . . . . . . . . 10
                     |
69 | 68 | adantl 468 |
. . . . . . . . 9
 
                     |
70 | | xp2nd 6824 |
. . . . . . . . . 10
                     |
71 | 70 | adantl 468 |
. . . . . . . . 9
 
                     |
72 | 1, 65, 66, 2, 3, 25, 67, 24, 69, 71 | xpcid 16074 |
. . . . . . . 8
 
                                                           |
73 | 64, 72 | eqtrd 2485 |
. . . . . . 7
 
                                                |
74 | | fvex 5875 |
. . . . . . . 8
             |
75 | | fvex 5875 |
. . . . . . . 8
             |
76 | 74, 75 | op1std 6803 |
. . . . . . 7
                                                              |
77 | 73, 76 | syl 17 |
. . . . . 6
 
                                     |
78 | 61, 77 | eqtrd 2485 |
. . . . 5
 
                                              |
79 | 1, 4, 5, 65, 66, 8, 58, 58 | 1stf2 16078 |
. . . . . 6
 
                   
  
       |
80 | 79 | fveq1d 5867 |
. . . . 5
 
                                                      |
81 | 50 | adantl 468 |
. . . . . 6
 
                                 |
82 | 81 | fveq2d 5869 |
. . . . 5
 
                                                 |
83 | 78, 80, 82 | 3eqtr4d 2495 |
. . . 4
 
                                                         |
84 | 28 | 3ad2ant1 1029 |
. . . . . . . 8
 
          
                                     
  |
85 | | simp21 1041 |
. . . . . . . 8
 
          
                                     
            |
86 | | simp22 1042 |
. . . . . . . 8
 
          
                                     
            |
87 | | simp23 1043 |
. . . . . . . 8
 
          
                                     
            |
88 | | simp3l 1036 |
. . . . . . . 8
 
          
                                     
         |
89 | | simp3r 1037 |
. . . . . . . 8
 
          
                                     
         |
90 | 4, 5, 26, 84, 85, 86, 87, 88, 89 | catcocl 15591 |
. . . . . . 7
 
          
                                     
       comp               |
91 | | fvres 5879 |
. . . . . . 7
        comp                                comp                 comp         |
92 | 90, 91 | syl 17 |
. . . . . 6
 
          
                                     
                   comp                 comp         |
93 | 1, 4, 5, 26, 85, 86, 87, 88, 89, 27 | xpcco1st 16069 |
. . . . . 6
 
          
                                     
          comp                          comp                |
94 | 92, 93 | eqtrd 2485 |
. . . . 5
 
          
                                     
                   comp                          comp                |
95 | 6 | 3ad2ant1 1029 |
. . . . . . 7
 
          
                                     
  |
96 | 7 | 3ad2ant1 1029 |
. . . . . . 7
 
          
                                     
  |
97 | 1, 4, 5, 95, 96, 8, 85, 87 | 1stf2 16078 |
. . . . . 6
 
          
                                     
                   |
98 | 97 | fveq1d 5867 |
. . . . 5
 
          
                                     
                  comp        
  
              comp         |
99 | 85, 50 | syl 17 |
. . . . . . . 8
 
          
                                     
                      |
100 | 86, 52 | syl 17 |
. . . . . . . 8
 
          
                                     
                      |
101 | 99, 100 | opeq12d 4174 |
. . . . . . 7
 
          
                                     
  
                                             |
102 | | fvres 5879 |
. . . . . . . 8
                                 |
103 | 87, 102 | syl 17 |
. . . . . . 7
 
          
                                     
                      |
104 | 101, 103 | oveq12d 6308 |
. . . . . 6
 
          
                                     
                                     comp    
                            comp          |
105 | 1, 4, 5, 95, 96, 8, 86, 87 | 1stf2 16078 |
. . . . . . . 8
 
          
                                     
                   |
106 | 105 | fveq1d 5867 |
. . . . . . 7
 
          
                                     
                           |
107 | | fvres 5879 |
. . . . . . . 8
         
                 |
108 | 89, 107 | syl 17 |
. . . . . . 7
 
          
                                     
                   |
109 | 106, 108 | eqtrd 2485 |
. . . . . 6
 
          
                                     
                  |
110 | 1, 4, 5, 95, 96, 8, 85, 86 | 1stf2 16078 |
. . . . . . . 8
 
          
                                     
                   |
111 | 110 | fveq1d 5867 |
. . . . . . 7
 
          
                                     
                           |
112 | | fvres 5879 |
. . . . . . . 8
         
                 |
113 | 88, 112 | syl 17 |
. . . . . . 7
 
          
                                     
                   |
114 | 111, 113 | eqtrd 2485 |
. . . . . 6
 
          
                                     
                  |
115 | 104, 109,
114 | oveq123d 6311 |
. . . . 5
 
          
                                     
                                                   comp    
                                                comp                |
116 | 94, 98, 115 | 3eqtr4d 2495 |
. . . 4
 
          
                                     
                  comp                                                          comp    
                               |
117 | 4, 2, 5, 23, 24, 25, 26, 27, 28, 6, 30, 37, 56, 83, 116 | isfuncd 15770 |
. . 3
              
        |
118 | | df-br 4403 |
. . 3
 
            
                             |
119 | 117, 118 | sylib 200 |
. 2
                        |
120 | 22, 119 | eqeltrd 2529 |
1
     |