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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 6549 | . . 3 | |
2 | elex 2697 | . . . . 5 | |
3 | 2 | 3ad2ant1 1002 | . . . 4 |
4 | elex 2697 | . . . . 5 | |
5 | 4 | 3ad2ant2 1003 | . . . 4 |
6 | fnovex 5804 | . . . 4 | |
7 | 1, 3, 5, 6 | mp3an2i 1320 | . . 3 |
8 | elex 2697 | . . . 4 | |
9 | 8 | 3ad2ant3 1004 | . . 3 |
10 | fnovex 5804 | . . 3 | |
11 | 1, 7, 9, 10 | mp3an2i 1320 | . 2 |
12 | xpexg 4653 | . . . 4 | |
13 | 12 | 3adant1 999 | . . 3 |
14 | fnovex 5804 | . . 3 | |
15 | 1, 3, 13, 14 | mp3an2i 1320 | . 2 |
16 | elmapi 6564 | . . . . . . . . . 10 | |
17 | 16 | ffvelrnda 5555 | . . . . . . . . 9 |
18 | elmapi 6564 | . . . . . . . . 9 | |
19 | 17, 18 | syl 14 | . . . . . . . 8 |
20 | 19 | ffvelrnda 5555 | . . . . . . 7 |
21 | 20 | an32s 557 | . . . . . 6 |
22 | 21 | ralrimiva 2505 | . . . . 5 |
23 | 22 | ralrimiva 2505 | . . . 4 |
24 | eqid 2139 | . . . . 5 | |
25 | 24 | fmpo 6099 | . . . 4 |
26 | 23, 25 | sylib 121 | . . 3 |
27 | simp1 981 | . . . 4 | |
28 | 27, 13 | elmapd 6556 | . . 3 |
29 | 26, 28 | syl5ibr 155 | . 2 |
30 | elmapi 6564 | . . . . . . . . 9 | |
31 | 30 | adantl 275 | . . . . . . . 8 |
32 | fovrn 5913 | . . . . . . . . . 10 | |
33 | 32 | 3expa 1181 | . . . . . . . . 9 |
34 | 33 | an32s 557 | . . . . . . . 8 |
35 | 31, 34 | sylanl1 399 | . . . . . . 7 |
36 | eqid 2139 | . . . . . . 7 | |
37 | 35, 36 | fmptd 5574 | . . . . . 6 |
38 | elmapg 6555 | . . . . . . . 8 | |
39 | 38 | 3adant3 1001 | . . . . . . 7 |
40 | 39 | ad2antrr 479 | . . . . . 6 |
41 | 37, 40 | mpbird 166 | . . . . 5 |
42 | eqid 2139 | . . . . 5 | |
43 | 41, 42 | fmptd 5574 | . . . 4 |
44 | 43 | ex 114 | . . 3 |
45 | simp3 983 | . . . 4 | |
46 | 7, 45 | elmapd 6556 | . . 3 |
47 | 44, 46 | sylibrd 168 | . 2 |
48 | elmapfn 6565 | . . . . . . . 8 | |
49 | 48 | ad2antll 482 | . . . . . . 7 |
50 | fnovim 5879 | . . . . . . 7 | |
51 | 49, 50 | syl 14 | . . . . . 6 |
52 | simp3 983 | . . . . . . . . . 10 | |
53 | 37 | adantlrl 473 | . . . . . . . . . . . 12 |
54 | 53 | 3adant2 1000 | . . . . . . . . . . 11 |
55 | simp1l2 1075 | . . . . . . . . . . 11 | |
56 | simp1l1 1074 | . . . . . . . . . . 11 | |
57 | fex2 5291 | . . . . . . . . . . 11 | |
58 | 54, 55, 56, 57 | syl3anc 1216 | . . . . . . . . . 10 |
59 | 42 | fvmpt2 5504 | . . . . . . . . . 10 |
60 | 52, 58, 59 | syl2anc 408 | . . . . . . . . 9 |
61 | 60 | fveq1d 5423 | . . . . . . . 8 |
62 | simp2 982 | . . . . . . . . 9 | |
63 | vex 2689 | . . . . . . . . . 10 | |
64 | vex 2689 | . . . . . . . . . 10 | |
65 | vex 2689 | . . . . . . . . . 10 | |
66 | ovexg 5805 | . . . . . . . . . 10 | |
67 | 63, 64, 65, 66 | mp3an 1315 | . . . . . . . . 9 |
68 | 36 | fvmpt2 5504 | . . . . . . . . 9 |
69 | 62, 67, 68 | sylancl 409 | . . . . . . . 8 |
70 | 61, 69 | eqtrd 2172 | . . . . . . 7 |
71 | 70 | mpoeq3dva 5835 | . . . . . 6 |
72 | 51, 71 | eqtr4d 2175 | . . . . 5 |
73 | eqid 2139 | . . . . . . 7 | |
74 | nfcv 2281 | . . . . . . . . . 10 | |
75 | nfmpt1 4021 | . . . . . . . . . 10 | |
76 | 74, 75 | nfmpt 4020 | . . . . . . . . 9 |
77 | 76 | nfeq2 2293 | . . . . . . . 8 |
78 | nfmpt1 4021 | . . . . . . . . . . . 12 | |
79 | 78 | nfeq2 2293 | . . . . . . . . . . 11 |
80 | fveq1 5420 | . . . . . . . . . . . . 13 | |
81 | 80 | fveq1d 5423 | . . . . . . . . . . . 12 |
82 | 81 | a1d 22 | . . . . . . . . . . 11 |
83 | 79, 82 | ralrimi 2503 | . . . . . . . . . 10 |
84 | eqid 2139 | . . . . . . . . . 10 | |
85 | 83, 84 | jctil 310 | . . . . . . . . 9 |
86 | 85 | a1d 22 | . . . . . . . 8 |
87 | 77, 86 | ralrimi 2503 | . . . . . . 7 |
88 | mpoeq123 5830 | . . . . . . 7 | |
89 | 73, 87, 88 | sylancr 410 | . . . . . 6 |
90 | 89 | eqeq2d 2151 | . . . . 5 |
91 | 72, 90 | syl5ibrcom 156 | . . . 4 |
92 | 16 | ad2antrl 481 | . . . . . . 7 |
93 | 92 | feqmptd 5474 | . . . . . 6 |
94 | simprl 520 | . . . . . . . . 9 | |
95 | 94, 19 | sylan 281 | . . . . . . . 8 |
96 | 95 | feqmptd 5474 | . . . . . . 7 |
97 | 96 | mpteq2dva 4018 | . . . . . 6 |
98 | 93, 97 | eqtrd 2172 | . . . . 5 |
99 | nfmpo2 5839 | . . . . . . . . 9 | |
100 | 99 | nfeq2 2293 | . . . . . . . 8 |
101 | eqidd 2140 | . . . . . . . . 9 | |
102 | nfmpo1 5838 | . . . . . . . . . . 11 | |
103 | 102 | nfeq2 2293 | . . . . . . . . . 10 |
104 | nfv 1508 | . . . . . . . . . 10 | |
105 | vex 2689 | . . . . . . . . . . . . . . 15 | |
106 | 105, 65 | fvex 5441 | . . . . . . . . . . . . . 14 |
107 | 106, 63 | fvex 5441 | . . . . . . . . . . . . 13 |
108 | 24 | ovmpt4g 5893 | . . . . . . . . . . . . 13 |
109 | 107, 108 | mp3an3 1304 | . . . . . . . . . . . 12 |
110 | oveq 5780 | . . . . . . . . . . . . 13 | |
111 | 110 | eqeq1d 2148 | . . . . . . . . . . . 12 |
112 | 109, 111 | syl5ibr 155 | . . . . . . . . . . 11 |
113 | 112 | expcomd 1417 | . . . . . . . . . 10 |
114 | 103, 104, 113 | ralrimd 2510 | . . . . . . . . 9 |
115 | mpteq12 4011 | . . . . . . . . 9 | |
116 | 101, 114, 115 | syl6an 1410 | . . . . . . . 8 |
117 | 100, 116 | ralrimi 2503 | . . . . . . 7 |
118 | mpteq12 4011 | . . . . . . 7 | |
119 | 84, 117, 118 | sylancr 410 | . . . . . 6 |
120 | 119 | eqeq2d 2151 | . . . . 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 6663 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wral 2416 cvv 2686 class class class wbr 3929 cmpt 3989 cxp 4537 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 cmpo 5776 cmap 6542 cen 6632 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-en 6635 |
This theorem is referenced by: (None) |
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