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Mirrors > Home > MPE Home > Th. List > xpsc0 | Structured version Visualization version Unicode version |
Description: The pair function maps to . (Contributed by Mario Carneiro, 14-Aug-2015.) |
Ref | Expression |
---|---|
xpsc0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsc 16217 | . . . 4 | |
2 | 1 | fveq1i 6192 | . . 3 |
3 | fnconstg 6093 | . . . 4 | |
4 | vex 3203 | . . . . . . . . . . . . 13 | |
5 | fvi 6255 | . . . . . . . . . . . . 13 | |
6 | 4, 5 | ax-mp 5 | . . . . . . . . . . . 12 |
7 | elsni 4194 | . . . . . . . . . . . . 13 | |
8 | 7 | fveq2d 6195 | . . . . . . . . . . . 12 |
9 | 6, 8 | syl5eqr 2670 | . . . . . . . . . . 11 |
10 | velsn 4193 | . . . . . . . . . . 11 | |
11 | 9, 10 | sylibr 224 | . . . . . . . . . 10 |
12 | 11 | ssriv 3607 | . . . . . . . . 9 |
13 | xpss2 5229 | . . . . . . . . 9 | |
14 | 12, 13 | ax-mp 5 | . . . . . . . 8 |
15 | 1on 7567 | . . . . . . . . . 10 | |
16 | 15 | elexi 3213 | . . . . . . . . 9 |
17 | fvex 6201 | . . . . . . . . 9 | |
18 | 16, 17 | xpsn 6407 | . . . . . . . 8 |
19 | 14, 18 | sseqtri 3637 | . . . . . . 7 |
20 | 16, 17 | funsn 5939 | . . . . . . 7 |
21 | funss 5907 | . . . . . . 7 | |
22 | 19, 20, 21 | mp2 9 | . . . . . 6 |
23 | funfn 5918 | . . . . . 6 | |
24 | 22, 23 | mpbi 220 | . . . . 5 |
25 | 24 | a1i 11 | . . . 4 |
26 | dmxpss 5565 | . . . . . . 7 | |
27 | sslin 3839 | . . . . . . 7 | |
28 | 26, 27 | ax-mp 5 | . . . . . 6 |
29 | 1n0 7575 | . . . . . . . 8 | |
30 | 29 | necomi 2848 | . . . . . . 7 |
31 | disjsn2 4247 | . . . . . . 7 | |
32 | 30, 31 | ax-mp 5 | . . . . . 6 |
33 | sseq0 3975 | . . . . . 6 | |
34 | 28, 32, 33 | mp2an 708 | . . . . 5 |
35 | 34 | a1i 11 | . . . 4 |
36 | 0ex 4790 | . . . . . 6 | |
37 | 36 | snid 4208 | . . . . 5 |
38 | 37 | a1i 11 | . . . 4 |
39 | fvun1 6269 | . . . 4 | |
40 | 3, 25, 35, 38, 39 | syl112anc 1330 | . . 3 |
41 | 2, 40 | syl5eq 2668 | . 2 |
42 | xpsng 6406 | . . . . 5 | |
43 | 42 | fveq1d 6193 | . . . 4 |
44 | fvsng 6447 | . . . 4 | |
45 | 43, 44 | eqtrd 2656 | . . 3 |
46 | 36, 45 | mpan 706 | . 2 |
47 | 41, 46 | eqtrd 2656 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 cun 3572 cin 3573 wss 3574 c0 3915 csn 4177 cop 4183 cid 5023 cxp 5112 ccnv 5113 cdm 5114 con0 5723 wfun 5882 wfn 5883 cfv 5888 (class class class)co 6650 c1o 7553 ccda 8989 |
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-3or 1038 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-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-ord 5726 df-on 5727 df-suc 5729 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-1o 7560 df-cda 8990 |
This theorem is referenced by: xpscfv 16222 xpsfeq 16224 xpsfrnel2 16225 xpsff1o 16228 xpsle 16241 dmdprdpr 18448 dprdpr 18449 xpstopnlem1 21612 xpstopnlem2 21614 xpsxmetlem 22184 xpsdsval 22186 xpsmet 22187 |
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