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Mirrors > Home > MPE Home > Th. List > mapxpen | Structured version Visualization version 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 | ovexd 6680 | . 2 | |
2 | ovexd 6680 | . 2 | |
3 | elmapi 7879 | . . . . . . . . . 10 | |
4 | 3 | ffvelrnda 6359 | . . . . . . . . 9 |
5 | elmapi 7879 | . . . . . . . . 9 | |
6 | 4, 5 | syl 17 | . . . . . . . 8 |
7 | 6 | ffvelrnda 6359 | . . . . . . 7 |
8 | 7 | an32s 846 | . . . . . 6 |
9 | 8 | ralrimiva 2966 | . . . . 5 |
10 | 9 | ralrimiva 2966 | . . . 4 |
11 | eqid 2622 | . . . . 5 | |
12 | 11 | fmpt2 7237 | . . . 4 |
13 | 10, 12 | sylib 208 | . . 3 |
14 | simp1 1061 | . . . 4 | |
15 | xpexg 6960 | . . . . 5 | |
16 | 15 | 3adant1 1079 | . . . 4 |
17 | elmapg 7870 | . . . 4 | |
18 | 14, 16, 17 | syl2anc 693 | . . 3 |
19 | 13, 18 | syl5ibr 236 | . 2 |
20 | elmapi 7879 | . . . . . . . . 9 | |
21 | 20 | adantl 482 | . . . . . . . 8 |
22 | fovrn 6804 | . . . . . . . . . 10 | |
23 | 22 | 3expa 1265 | . . . . . . . . 9 |
24 | 23 | an32s 846 | . . . . . . . 8 |
25 | 21, 24 | sylanl1 682 | . . . . . . 7 |
26 | eqid 2622 | . . . . . . 7 | |
27 | 25, 26 | fmptd 6385 | . . . . . 6 |
28 | elmapg 7870 | . . . . . . . 8 | |
29 | 28 | 3adant3 1081 | . . . . . . 7 |
30 | 29 | ad2antrr 762 | . . . . . 6 |
31 | 27, 30 | mpbird 247 | . . . . 5 |
32 | eqid 2622 | . . . . 5 | |
33 | 31, 32 | fmptd 6385 | . . . 4 |
34 | 33 | ex 450 | . . 3 |
35 | ovex 6678 | . . . 4 | |
36 | simp3 1063 | . . . 4 | |
37 | elmapg 7870 | . . . 4 | |
38 | 35, 36, 37 | sylancr 695 | . . 3 |
39 | 34, 38 | sylibrd 249 | . 2 |
40 | elmapfn 7880 | . . . . . . . 8 | |
41 | 40 | ad2antll 765 | . . . . . . 7 |
42 | fnov 6768 | . . . . . . 7 | |
43 | 41, 42 | sylib 208 | . . . . . 6 |
44 | simp3 1063 | . . . . . . . . . 10 | |
45 | 27 | adantlrl 756 | . . . . . . . . . . . 12 |
46 | 45 | 3adant2 1080 | . . . . . . . . . . 11 |
47 | simp1l2 1155 | . . . . . . . . . . 11 | |
48 | simp1l1 1154 | . . . . . . . . . . 11 | |
49 | fex2 7121 | . . . . . . . . . . 11 | |
50 | 46, 47, 48, 49 | syl3anc 1326 | . . . . . . . . . 10 |
51 | 32 | fvmpt2 6291 | . . . . . . . . . 10 |
52 | 44, 50, 51 | syl2anc 693 | . . . . . . . . 9 |
53 | 52 | fveq1d 6193 | . . . . . . . 8 |
54 | simp2 1062 | . . . . . . . . 9 | |
55 | ovex 6678 | . . . . . . . . 9 | |
56 | 26 | fvmpt2 6291 | . . . . . . . . 9 |
57 | 54, 55, 56 | sylancl 694 | . . . . . . . 8 |
58 | 53, 57 | eqtrd 2656 | . . . . . . 7 |
59 | 58 | mpt2eq3dva 6719 | . . . . . 6 |
60 | 43, 59 | eqtr4d 2659 | . . . . 5 |
61 | eqid 2622 | . . . . . . 7 | |
62 | nfcv 2764 | . . . . . . . . . 10 | |
63 | nfmpt1 4747 | . . . . . . . . . 10 | |
64 | 62, 63 | nfmpt 4746 | . . . . . . . . 9 |
65 | 64 | nfeq2 2780 | . . . . . . . 8 |
66 | nfmpt1 4747 | . . . . . . . . . . . 12 | |
67 | 66 | nfeq2 2780 | . . . . . . . . . . 11 |
68 | fveq1 6190 | . . . . . . . . . . . . 13 | |
69 | 68 | fveq1d 6193 | . . . . . . . . . . . 12 |
70 | 69 | a1d 25 | . . . . . . . . . . 11 |
71 | 67, 70 | ralrimi 2957 | . . . . . . . . . 10 |
72 | eqid 2622 | . . . . . . . . . 10 | |
73 | 71, 72 | jctil 560 | . . . . . . . . 9 |
74 | 73 | a1d 25 | . . . . . . . 8 |
75 | 65, 74 | ralrimi 2957 | . . . . . . 7 |
76 | mpt2eq123 6714 | . . . . . . 7 | |
77 | 61, 75, 76 | sylancr 695 | . . . . . 6 |
78 | 77 | eqeq2d 2632 | . . . . 5 |
79 | 60, 78 | syl5ibrcom 237 | . . . 4 |
80 | 3 | ad2antrl 764 | . . . . . . 7 |
81 | 80 | feqmptd 6249 | . . . . . 6 |
82 | simprl 794 | . . . . . . . . 9 | |
83 | 82, 6 | sylan 488 | . . . . . . . 8 |
84 | 83 | feqmptd 6249 | . . . . . . 7 |
85 | 84 | mpteq2dva 4744 | . . . . . 6 |
86 | 81, 85 | eqtrd 2656 | . . . . 5 |
87 | nfmpt22 6723 | . . . . . . . . 9 | |
88 | 87 | nfeq2 2780 | . . . . . . . 8 |
89 | eqidd 2623 | . . . . . . . . 9 | |
90 | nfmpt21 6722 | . . . . . . . . . . 11 | |
91 | 90 | nfeq2 2780 | . . . . . . . . . 10 |
92 | nfv 1843 | . . . . . . . . . 10 | |
93 | fvex 6201 | . . . . . . . . . . . . 13 | |
94 | 11 | ovmpt4g 6783 | . . . . . . . . . . . . 13 |
95 | 93, 94 | mp3an3 1413 | . . . . . . . . . . . 12 |
96 | oveq 6656 | . . . . . . . . . . . . 13 | |
97 | 96 | eqeq1d 2624 | . . . . . . . . . . . 12 |
98 | 95, 97 | syl5ibr 236 | . . . . . . . . . . 11 |
99 | 98 | expcomd 454 | . . . . . . . . . 10 |
100 | 91, 92, 99 | ralrimd 2959 | . . . . . . . . 9 |
101 | mpteq12 4736 | . . . . . . . . 9 | |
102 | 89, 100, 101 | syl6an 568 | . . . . . . . 8 |
103 | 88, 102 | ralrimi 2957 | . . . . . . 7 |
104 | mpteq12 4736 | . . . . . . 7 | |
105 | 72, 103, 104 | sylancr 695 | . . . . . 6 |
106 | 105 | eqeq2d 2632 | . . . . 5 |
107 | 86, 106 | syl5ibrcom 237 | . . . 4 |
108 | 79, 107 | impbid 202 | . . 3 |
109 | 108 | ex 450 | . 2 |
110 | 1, 2, 19, 39, 109 | en3d 7992 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 cvv 3200 class class class wbr 4653 cmpt 4729 cxp 5112 wfn 5883 wf 5884 cfv 5888 (class class class)co 6650 cmpt2 6652 cmap 7857 cen 7952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 df-en 7956 |
This theorem is referenced by: mappwen 8935 cfpwsdom 9406 rpnnen 14956 rexpen 14957 enrelmap 38291 |
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