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Mirrors > Home > ILE Home > Th. List > mapxpen | Unicode version |
Description: Equinumerosity law for double set exponentiation. Proposition 10.45 of [TakeutiZaring] p. 96. (Contributed by NM, 21-Feb-2004.) (Revised by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
mapxpen |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmap 6612 | . . 3 | |
2 | elex 2732 | . . . . 5 | |
3 | 2 | 3ad2ant1 1007 | . . . 4 |
4 | elex 2732 | . . . . 5 | |
5 | 4 | 3ad2ant2 1008 | . . . 4 |
6 | fnovex 5866 | . . . 4 | |
7 | 1, 3, 5, 6 | mp3an2i 1331 | . . 3 |
8 | elex 2732 | . . . 4 | |
9 | 8 | 3ad2ant3 1009 | . . 3 |
10 | fnovex 5866 | . . 3 | |
11 | 1, 7, 9, 10 | mp3an2i 1331 | . 2 |
12 | xpexg 4712 | . . . 4 | |
13 | 12 | 3adant1 1004 | . . 3 |
14 | fnovex 5866 | . . 3 | |
15 | 1, 3, 13, 14 | mp3an2i 1331 | . 2 |
16 | elmapi 6627 | . . . . . . . . . 10 | |
17 | 16 | ffvelrnda 5614 | . . . . . . . . 9 |
18 | elmapi 6627 | . . . . . . . . 9 | |
19 | 17, 18 | syl 14 | . . . . . . . 8 |
20 | 19 | ffvelrnda 5614 | . . . . . . 7 |
21 | 20 | an32s 558 | . . . . . 6 |
22 | 21 | ralrimiva 2537 | . . . . 5 |
23 | 22 | ralrimiva 2537 | . . . 4 |
24 | eqid 2164 | . . . . 5 | |
25 | 24 | fmpo 6161 | . . . 4 |
26 | 23, 25 | sylib 121 | . . 3 |
27 | simp1 986 | . . . 4 | |
28 | 27, 13 | elmapd 6619 | . . 3 |
29 | 26, 28 | syl5ibr 155 | . 2 |
30 | elmapi 6627 | . . . . . . . . 9 | |
31 | 30 | adantl 275 | . . . . . . . 8 |
32 | fovrn 5975 | . . . . . . . . . 10 | |
33 | 32 | 3expa 1192 | . . . . . . . . 9 |
34 | 33 | an32s 558 | . . . . . . . 8 |
35 | 31, 34 | sylanl1 400 | . . . . . . 7 |
36 | eqid 2164 | . . . . . . 7 | |
37 | 35, 36 | fmptd 5633 | . . . . . 6 |
38 | elmapg 6618 | . . . . . . . 8 | |
39 | 38 | 3adant3 1006 | . . . . . . 7 |
40 | 39 | ad2antrr 480 | . . . . . 6 |
41 | 37, 40 | mpbird 166 | . . . . 5 |
42 | eqid 2164 | . . . . 5 | |
43 | 41, 42 | fmptd 5633 | . . . 4 |
44 | 43 | ex 114 | . . 3 |
45 | simp3 988 | . . . 4 | |
46 | 7, 45 | elmapd 6619 | . . 3 |
47 | 44, 46 | sylibrd 168 | . 2 |
48 | elmapfn 6628 | . . . . . . . 8 | |
49 | 48 | ad2antll 483 | . . . . . . 7 |
50 | fnovim 5941 | . . . . . . 7 | |
51 | 49, 50 | syl 14 | . . . . . 6 |
52 | simp3 988 | . . . . . . . . . 10 | |
53 | 37 | adantlrl 474 | . . . . . . . . . . . 12 |
54 | 53 | 3adant2 1005 | . . . . . . . . . . 11 |
55 | simp1l2 1080 | . . . . . . . . . . 11 | |
56 | simp1l1 1079 | . . . . . . . . . . 11 | |
57 | fex2 5350 | . . . . . . . . . . 11 | |
58 | 54, 55, 56, 57 | syl3anc 1227 | . . . . . . . . . 10 |
59 | 42 | fvmpt2 5563 | . . . . . . . . . 10 |
60 | 52, 58, 59 | syl2anc 409 | . . . . . . . . 9 |
61 | 60 | fveq1d 5482 | . . . . . . . 8 |
62 | simp2 987 | . . . . . . . . 9 | |
63 | vex 2724 | . . . . . . . . . 10 | |
64 | vex 2724 | . . . . . . . . . 10 | |
65 | vex 2724 | . . . . . . . . . 10 | |
66 | ovexg 5867 | . . . . . . . . . 10 | |
67 | 63, 64, 65, 66 | mp3an 1326 | . . . . . . . . 9 |
68 | 36 | fvmpt2 5563 | . . . . . . . . 9 |
69 | 62, 67, 68 | sylancl 410 | . . . . . . . 8 |
70 | 61, 69 | eqtrd 2197 | . . . . . . 7 |
71 | 70 | mpoeq3dva 5897 | . . . . . 6 |
72 | 51, 71 | eqtr4d 2200 | . . . . 5 |
73 | eqid 2164 | . . . . . . 7 | |
74 | nfcv 2306 | . . . . . . . . . 10 | |
75 | nfmpt1 4069 | . . . . . . . . . 10 | |
76 | 74, 75 | nfmpt 4068 | . . . . . . . . 9 |
77 | 76 | nfeq2 2318 | . . . . . . . 8 |
78 | nfmpt1 4069 | . . . . . . . . . . . 12 | |
79 | 78 | nfeq2 2318 | . . . . . . . . . . 11 |
80 | fveq1 5479 | . . . . . . . . . . . . 13 | |
81 | 80 | fveq1d 5482 | . . . . . . . . . . . 12 |
82 | 81 | a1d 22 | . . . . . . . . . . 11 |
83 | 79, 82 | ralrimi 2535 | . . . . . . . . . 10 |
84 | eqid 2164 | . . . . . . . . . 10 | |
85 | 83, 84 | jctil 310 | . . . . . . . . 9 |
86 | 85 | a1d 22 | . . . . . . . 8 |
87 | 77, 86 | ralrimi 2535 | . . . . . . 7 |
88 | mpoeq123 5892 | . . . . . . 7 | |
89 | 73, 87, 88 | sylancr 411 | . . . . . 6 |
90 | 89 | eqeq2d 2176 | . . . . 5 |
91 | 72, 90 | syl5ibrcom 156 | . . . 4 |
92 | 16 | ad2antrl 482 | . . . . . . 7 |
93 | 92 | feqmptd 5533 | . . . . . 6 |
94 | simprl 521 | . . . . . . . . 9 | |
95 | 94, 19 | sylan 281 | . . . . . . . 8 |
96 | 95 | feqmptd 5533 | . . . . . . 7 |
97 | 96 | mpteq2dva 4066 | . . . . . 6 |
98 | 93, 97 | eqtrd 2197 | . . . . 5 |
99 | nfmpo2 5901 | . . . . . . . . 9 | |
100 | 99 | nfeq2 2318 | . . . . . . . 8 |
101 | eqidd 2165 | . . . . . . . . 9 | |
102 | nfmpo1 5900 | . . . . . . . . . . 11 | |
103 | 102 | nfeq2 2318 | . . . . . . . . . 10 |
104 | nfv 1515 | . . . . . . . . . 10 | |
105 | vex 2724 | . . . . . . . . . . . . . . 15 | |
106 | 105, 65 | fvex 5500 | . . . . . . . . . . . . . 14 |
107 | 106, 63 | fvex 5500 | . . . . . . . . . . . . 13 |
108 | 24 | ovmpt4g 5955 | . . . . . . . . . . . . 13 |
109 | 107, 108 | mp3an3 1315 | . . . . . . . . . . . 12 |
110 | oveq 5842 | . . . . . . . . . . . . 13 | |
111 | 110 | eqeq1d 2173 | . . . . . . . . . . . 12 |
112 | 109, 111 | syl5ibr 155 | . . . . . . . . . . 11 |
113 | 112 | expcomd 1428 | . . . . . . . . . 10 |
114 | 103, 104, 113 | ralrimd 2542 | . . . . . . . . 9 |
115 | mpteq12 4059 | . . . . . . . . 9 | |
116 | 101, 114, 115 | syl6an 1421 | . . . . . . . 8 |
117 | 100, 116 | ralrimi 2535 | . . . . . . 7 |
118 | mpteq12 4059 | . . . . . . 7 | |
119 | 84, 117, 118 | sylancr 411 | . . . . . 6 |
120 | 119 | eqeq2d 2176 | . . . . 5 |
121 | 98, 120 | syl5ibrcom 156 | . . . 4 |
122 | 91, 121 | impbid 128 | . . 3 |
123 | 122 | ex 114 | . 2 |
124 | 11, 15, 29, 47, 123 | en3d 6726 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wral 2442 cvv 2721 class class class wbr 3976 cmpt 4037 cxp 4596 wfn 5177 wf 5178 cfv 5182 (class class class)co 5836 cmpo 5838 cmap 6605 cen 6695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-map 6607 df-en 6698 |
This theorem is referenced by: (None) |
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