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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 6629 | . . 3 | |
2 | elex 2741 | . . . . 5 | |
3 | 2 | 3ad2ant1 1013 | . . . 4 |
4 | elex 2741 | . . . . 5 | |
5 | 4 | 3ad2ant2 1014 | . . . 4 |
6 | fnovex 5883 | . . . 4 | |
7 | 1, 3, 5, 6 | mp3an2i 1337 | . . 3 |
8 | elex 2741 | . . . 4 | |
9 | 8 | 3ad2ant3 1015 | . . 3 |
10 | fnovex 5883 | . . 3 | |
11 | 1, 7, 9, 10 | mp3an2i 1337 | . 2 |
12 | xpexg 4723 | . . . 4 | |
13 | 12 | 3adant1 1010 | . . 3 |
14 | fnovex 5883 | . . 3 | |
15 | 1, 3, 13, 14 | mp3an2i 1337 | . 2 |
16 | elmapi 6644 | . . . . . . . . . 10 | |
17 | 16 | ffvelrnda 5628 | . . . . . . . . 9 |
18 | elmapi 6644 | . . . . . . . . 9 | |
19 | 17, 18 | syl 14 | . . . . . . . 8 |
20 | 19 | ffvelrnda 5628 | . . . . . . 7 |
21 | 20 | an32s 563 | . . . . . 6 |
22 | 21 | ralrimiva 2543 | . . . . 5 |
23 | 22 | ralrimiva 2543 | . . . 4 |
24 | eqid 2170 | . . . . 5 | |
25 | 24 | fmpo 6177 | . . . 4 |
26 | 23, 25 | sylib 121 | . . 3 |
27 | simp1 992 | . . . 4 | |
28 | 27, 13 | elmapd 6636 | . . 3 |
29 | 26, 28 | syl5ibr 155 | . 2 |
30 | elmapi 6644 | . . . . . . . . 9 | |
31 | 30 | adantl 275 | . . . . . . . 8 |
32 | fovrn 5992 | . . . . . . . . . 10 | |
33 | 32 | 3expa 1198 | . . . . . . . . 9 |
34 | 33 | an32s 563 | . . . . . . . 8 |
35 | 31, 34 | sylanl1 400 | . . . . . . 7 |
36 | eqid 2170 | . . . . . . 7 | |
37 | 35, 36 | fmptd 5647 | . . . . . 6 |
38 | elmapg 6635 | . . . . . . . 8 | |
39 | 38 | 3adant3 1012 | . . . . . . 7 |
40 | 39 | ad2antrr 485 | . . . . . 6 |
41 | 37, 40 | mpbird 166 | . . . . 5 |
42 | eqid 2170 | . . . . 5 | |
43 | 41, 42 | fmptd 5647 | . . . 4 |
44 | 43 | ex 114 | . . 3 |
45 | simp3 994 | . . . 4 | |
46 | 7, 45 | elmapd 6636 | . . 3 |
47 | 44, 46 | sylibrd 168 | . 2 |
48 | elmapfn 6645 | . . . . . . . 8 | |
49 | 48 | ad2antll 488 | . . . . . . 7 |
50 | fnovim 5958 | . . . . . . 7 | |
51 | 49, 50 | syl 14 | . . . . . 6 |
52 | simp3 994 | . . . . . . . . . 10 | |
53 | 37 | adantlrl 479 | . . . . . . . . . . . 12 |
54 | 53 | 3adant2 1011 | . . . . . . . . . . 11 |
55 | simp1l2 1086 | . . . . . . . . . . 11 | |
56 | simp1l1 1085 | . . . . . . . . . . 11 | |
57 | fex2 5364 | . . . . . . . . . . 11 | |
58 | 54, 55, 56, 57 | syl3anc 1233 | . . . . . . . . . 10 |
59 | 42 | fvmpt2 5577 | . . . . . . . . . 10 |
60 | 52, 58, 59 | syl2anc 409 | . . . . . . . . 9 |
61 | 60 | fveq1d 5496 | . . . . . . . 8 |
62 | simp2 993 | . . . . . . . . 9 | |
63 | vex 2733 | . . . . . . . . . 10 | |
64 | vex 2733 | . . . . . . . . . 10 | |
65 | vex 2733 | . . . . . . . . . 10 | |
66 | ovexg 5884 | . . . . . . . . . 10 | |
67 | 63, 64, 65, 66 | mp3an 1332 | . . . . . . . . 9 |
68 | 36 | fvmpt2 5577 | . . . . . . . . 9 |
69 | 62, 67, 68 | sylancl 411 | . . . . . . . 8 |
70 | 61, 69 | eqtrd 2203 | . . . . . . 7 |
71 | 70 | mpoeq3dva 5914 | . . . . . 6 |
72 | 51, 71 | eqtr4d 2206 | . . . . 5 |
73 | eqid 2170 | . . . . . . 7 | |
74 | nfcv 2312 | . . . . . . . . . 10 | |
75 | nfmpt1 4080 | . . . . . . . . . 10 | |
76 | 74, 75 | nfmpt 4079 | . . . . . . . . 9 |
77 | 76 | nfeq2 2324 | . . . . . . . 8 |
78 | nfmpt1 4080 | . . . . . . . . . . . 12 | |
79 | 78 | nfeq2 2324 | . . . . . . . . . . 11 |
80 | fveq1 5493 | . . . . . . . . . . . . 13 | |
81 | 80 | fveq1d 5496 | . . . . . . . . . . . 12 |
82 | 81 | a1d 22 | . . . . . . . . . . 11 |
83 | 79, 82 | ralrimi 2541 | . . . . . . . . . 10 |
84 | eqid 2170 | . . . . . . . . . 10 | |
85 | 83, 84 | jctil 310 | . . . . . . . . 9 |
86 | 85 | a1d 22 | . . . . . . . 8 |
87 | 77, 86 | ralrimi 2541 | . . . . . . 7 |
88 | mpoeq123 5909 | . . . . . . 7 | |
89 | 73, 87, 88 | sylancr 412 | . . . . . 6 |
90 | 89 | eqeq2d 2182 | . . . . 5 |
91 | 72, 90 | syl5ibrcom 156 | . . . 4 |
92 | 16 | ad2antrl 487 | . . . . . . 7 |
93 | 92 | feqmptd 5547 | . . . . . 6 |
94 | simprl 526 | . . . . . . . . 9 | |
95 | 94, 19 | sylan 281 | . . . . . . . 8 |
96 | 95 | feqmptd 5547 | . . . . . . 7 |
97 | 96 | mpteq2dva 4077 | . . . . . 6 |
98 | 93, 97 | eqtrd 2203 | . . . . 5 |
99 | nfmpo2 5918 | . . . . . . . . 9 | |
100 | 99 | nfeq2 2324 | . . . . . . . 8 |
101 | eqidd 2171 | . . . . . . . . 9 | |
102 | nfmpo1 5917 | . . . . . . . . . . 11 | |
103 | 102 | nfeq2 2324 | . . . . . . . . . 10 |
104 | nfv 1521 | . . . . . . . . . 10 | |
105 | vex 2733 | . . . . . . . . . . . . . . 15 | |
106 | 105, 65 | fvex 5514 | . . . . . . . . . . . . . 14 |
107 | 106, 63 | fvex 5514 | . . . . . . . . . . . . 13 |
108 | 24 | ovmpt4g 5972 | . . . . . . . . . . . . 13 |
109 | 107, 108 | mp3an3 1321 | . . . . . . . . . . . 12 |
110 | oveq 5856 | . . . . . . . . . . . . 13 | |
111 | 110 | eqeq1d 2179 | . . . . . . . . . . . 12 |
112 | 109, 111 | syl5ibr 155 | . . . . . . . . . . 11 |
113 | 112 | expcomd 1434 | . . . . . . . . . 10 |
114 | 103, 104, 113 | ralrimd 2548 | . . . . . . . . 9 |
115 | mpteq12 4070 | . . . . . . . . 9 | |
116 | 101, 114, 115 | syl6an 1427 | . . . . . . . 8 |
117 | 100, 116 | ralrimi 2541 | . . . . . . 7 |
118 | mpteq12 4070 | . . . . . . 7 | |
119 | 84, 117, 118 | sylancr 412 | . . . . . 6 |
120 | 119 | eqeq2d 2182 | . . . . 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 6743 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wral 2448 cvv 2730 class class class wbr 3987 cmpt 4048 cxp 4607 wfn 5191 wf 5192 cfv 5196 (class class class)co 5850 cmpo 5852 cmap 6622 cen 6712 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-map 6624 df-en 6715 |
This theorem is referenced by: (None) |
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