Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > resf1o | Structured version Visualization version Unicode version |
Description: Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.) |
Ref | Expression |
---|---|
resf1o.1 | |
resf1o.2 |
Ref | Expression |
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resf1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resf1o.2 | . 2 | |
2 | resexg 5442 | . . 3 | |
3 | 2 | adantl 482 | . 2 |
4 | simpr 477 | . . . 4 | |
5 | difexg 4808 | . . . . . . 7 | |
6 | 5 | 3ad2ant1 1082 | . . . . . 6 |
7 | snex 4908 | . . . . . 6 | |
8 | xpexg 6960 | . . . . . 6 | |
9 | 6, 7, 8 | sylancl 694 | . . . . 5 |
10 | 9 | adantr 481 | . . . 4 |
11 | unexg 6959 | . . . 4 | |
12 | 4, 10, 11 | syl2anc 693 | . . 3 |
13 | 12 | adantlr 751 | . 2 |
14 | resf1o.1 | . . . . 5 | |
15 | 14 | rabeq2i 3197 | . . . 4 |
16 | 15 | anbi1i 731 | . . 3 |
17 | simprr 796 | . . . . . 6 | |
18 | simprll 802 | . . . . . . . . 9 | |
19 | elmapi 7879 | . . . . . . . . 9 | |
20 | 18, 19 | syl 17 | . . . . . . . 8 |
21 | simp3 1063 | . . . . . . . . 9 | |
22 | 21 | ad2antrr 762 | . . . . . . . 8 |
23 | 20, 22 | fssresd 6071 | . . . . . . 7 |
24 | simp2 1062 | . . . . . . . . 9 | |
25 | simp1 1061 | . . . . . . . . . 10 | |
26 | 25, 21 | ssexd 4805 | . . . . . . . . 9 |
27 | elmapg 7870 | . . . . . . . . 9 | |
28 | 24, 26, 27 | syl2anc 693 | . . . . . . . 8 |
29 | 28 | ad2antrr 762 | . . . . . . 7 |
30 | 23, 29 | mpbird 247 | . . . . . 6 |
31 | 17, 30 | eqeltrd 2701 | . . . . 5 |
32 | undif 4049 | . . . . . . . . . . 11 | |
33 | 32 | biimpi 206 | . . . . . . . . . 10 |
34 | 33 | reseq2d 5396 | . . . . . . . . 9 |
35 | 22, 34 | syl 17 | . . . . . . . 8 |
36 | ffn 6045 | . . . . . . . . 9 | |
37 | fnresdm 6000 | . . . . . . . . 9 | |
38 | 20, 36, 37 | 3syl 18 | . . . . . . . 8 |
39 | 35, 38 | eqtr2d 2657 | . . . . . . 7 |
40 | resundi 5410 | . . . . . . 7 | |
41 | 39, 40 | syl6eq 2672 | . . . . . 6 |
42 | 17 | eqcomd 2628 | . . . . . . 7 |
43 | simprlr 803 | . . . . . . . . 9 | |
44 | 25 | ad2antrr 762 | . . . . . . . . . 10 |
45 | simplr 792 | . . . . . . . . . 10 | |
46 | eqid 2622 | . . . . . . . . . . 11 | |
47 | 46 | ffs2 29503 | . . . . . . . . . 10 supp |
48 | 44, 45, 20, 47 | syl3anc 1326 | . . . . . . . . 9 supp |
49 | sseqin2 3817 | . . . . . . . . . . 11 | |
50 | 49 | biimpi 206 | . . . . . . . . . 10 |
51 | 22, 50 | syl 17 | . . . . . . . . 9 |
52 | 43, 48, 51 | 3sstr4d 3648 | . . . . . . . 8 supp |
53 | simpl 473 | . . . . . . . . . . . 12 | |
54 | 53, 19, 36 | 3syl 18 | . . . . . . . . . . 11 |
55 | inundif 4046 | . . . . . . . . . . . 12 | |
56 | 55 | fneq2i 5986 | . . . . . . . . . . 11 |
57 | 54, 56 | sylibr 224 | . . . . . . . . . 10 |
58 | vex 3203 | . . . . . . . . . . 11 | |
59 | 58 | a1i 11 | . . . . . . . . . 10 |
60 | simpr 477 | . . . . . . . . . 10 | |
61 | inindif 29353 | . . . . . . . . . . 11 | |
62 | 61 | a1i 11 | . . . . . . . . . 10 |
63 | fnsuppres 7322 | . . . . . . . . . 10 supp | |
64 | 57, 59, 60, 62, 63 | syl121anc 1331 | . . . . . . . . 9 supp |
65 | 18, 45, 64 | syl2anc 693 | . . . . . . . 8 supp |
66 | 52, 65 | mpbid 222 | . . . . . . 7 |
67 | 42, 66 | uneq12d 3768 | . . . . . 6 |
68 | 41, 67 | eqtrd 2656 | . . . . 5 |
69 | 31, 68 | jca 554 | . . . 4 |
70 | 24 | ad2antrr 762 | . . . . . 6 |
71 | 25 | ad2antrr 762 | . . . . . 6 |
72 | elmapi 7879 | . . . . . . . . 9 | |
73 | 72 | ad2antrl 764 | . . . . . . . 8 |
74 | simplr 792 | . . . . . . . . 9 | |
75 | fconst6g 6094 | . . . . . . . . 9 | |
76 | 74, 75 | syl 17 | . . . . . . . 8 |
77 | disjdif 4040 | . . . . . . . . 9 | |
78 | 77 | a1i 11 | . . . . . . . 8 |
79 | fun2 6067 | . . . . . . . 8 | |
80 | 73, 76, 78, 79 | syl21anc 1325 | . . . . . . 7 |
81 | simprr 796 | . . . . . . . . 9 | |
82 | 81 | eqcomd 2628 | . . . . . . . 8 |
83 | 21 | ad2antrr 762 | . . . . . . . . 9 |
84 | 83, 33 | syl 17 | . . . . . . . 8 |
85 | 82, 84 | feq12d 6033 | . . . . . . 7 |
86 | 80, 85 | mpbid 222 | . . . . . 6 |
87 | elmapg 7870 | . . . . . . 7 | |
88 | 87 | biimpar 502 | . . . . . 6 |
89 | 70, 71, 86, 88 | syl21anc 1325 | . . . . 5 |
90 | 71, 74, 86, 47 | syl3anc 1326 | . . . . . 6 supp |
91 | 81 | adantr 481 | . . . . . . . . 9 |
92 | 91 | fveq1d 6193 | . . . . . . . 8 |
93 | 73 | adantr 481 | . . . . . . . . . 10 |
94 | ffn 6045 | . . . . . . . . . 10 | |
95 | 93, 94 | syl 17 | . . . . . . . . 9 |
96 | fconstg 6092 | . . . . . . . . . . 11 | |
97 | 96 | ad3antlr 767 | . . . . . . . . . 10 |
98 | ffn 6045 | . . . . . . . . . 10 | |
99 | 97, 98 | syl 17 | . . . . . . . . 9 |
100 | 77 | a1i 11 | . . . . . . . . 9 |
101 | simpr 477 | . . . . . . . . 9 | |
102 | fvun2 6270 | . . . . . . . . 9 | |
103 | 95, 99, 100, 101, 102 | syl112anc 1330 | . . . . . . . 8 |
104 | fvconst 6431 | . . . . . . . . 9 | |
105 | 97, 101, 104 | syl2anc 693 | . . . . . . . 8 |
106 | 92, 103, 105 | 3eqtrd 2660 | . . . . . . 7 |
107 | 86, 106 | suppss 7325 | . . . . . 6 supp |
108 | 90, 107 | eqsstr3d 3640 | . . . . 5 |
109 | 81 | reseq1d 5395 | . . . . . 6 |
110 | res0 5400 | . . . . . . . . . 10 | |
111 | res0 5400 | . . . . . . . . . 10 | |
112 | 110, 111 | eqtr4i 2647 | . . . . . . . . 9 |
113 | 77 | reseq2i 5393 | . . . . . . . . 9 |
114 | 77 | reseq2i 5393 | . . . . . . . . 9 |
115 | 112, 113, 114 | 3eqtr4ri 2655 | . . . . . . . 8 |
116 | 115 | a1i 11 | . . . . . . 7 |
117 | fresaunres1 6077 | . . . . . . 7 | |
118 | 73, 76, 116, 117 | syl3anc 1326 | . . . . . 6 |
119 | 109, 118 | eqtr2d 2657 | . . . . 5 |
120 | 89, 108, 119 | jca31 557 | . . . 4 |
121 | 69, 120 | impbida 877 | . . 3 |
122 | 16, 121 | syl5bb 272 | . 2 |
123 | 1, 3, 13, 122 | f1od 6885 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 crab 2916 cvv 3200 cdif 3571 cun 3572 cin 3573 wss 3574 c0 3915 csn 4177 cmpt 4729 cxp 5112 ccnv 5113 cres 5116 cima 5117 wfn 5883 wf 5884 wf1o 5887 cfv 5888 (class class class)co 6650 supp csupp 7295 cmap 7857 |
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-rep 4771 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-reu 2919 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-supp 7296 df-map 7859 |
This theorem is referenced by: eulerpartgbij 30434 |
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