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Mirrors > Home > MPE Home > Th. List > fconst5 | Structured version Visualization version Unicode version |
Description: Two ways to express that a function is constant. (Contributed by NM, 27-Nov-2007.) |
Ref | Expression |
---|---|
fconst5 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rneq 5351 | . . . 4 | |
2 | rnxp 5564 | . . . . 5 | |
3 | 2 | eqeq2d 2632 | . . . 4 |
4 | 1, 3 | syl5ib 234 | . . 3 |
5 | 4 | adantl 482 | . 2 |
6 | df-fo 5894 | . . . . . . 7 | |
7 | fof 6115 | . . . . . . 7 | |
8 | 6, 7 | sylbir 225 | . . . . . 6 |
9 | fconst2g 6468 | . . . . . 6 | |
10 | 8, 9 | syl5ib 234 | . . . . 5 |
11 | 10 | expd 452 | . . . 4 |
12 | 11 | adantrd 484 | . . 3 |
13 | fnrel 5989 | . . . . 5 | |
14 | snprc 4253 | . . . . . 6 | |
15 | relrn0 5383 | . . . . . . . . . 10 | |
16 | 15 | biimprd 238 | . . . . . . . . 9 |
17 | 16 | adantl 482 | . . . . . . . 8 |
18 | eqeq2 2633 | . . . . . . . . 9 | |
19 | 18 | adantr 481 | . . . . . . . 8 |
20 | xpeq2 5129 | . . . . . . . . . . 11 | |
21 | xp0 5552 | . . . . . . . . . . 11 | |
22 | 20, 21 | syl6eq 2672 | . . . . . . . . . 10 |
23 | 22 | eqeq2d 2632 | . . . . . . . . 9 |
24 | 23 | adantr 481 | . . . . . . . 8 |
25 | 17, 19, 24 | 3imtr4d 283 | . . . . . . 7 |
26 | 25 | ex 450 | . . . . . 6 |
27 | 14, 26 | sylbi 207 | . . . . 5 |
28 | 13, 27 | syl5 34 | . . . 4 |
29 | 28 | adantrd 484 | . . 3 |
30 | 12, 29 | pm2.61i 176 | . 2 |
31 | 5, 30 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 c0 3915 csn 4177 cxp 5112 crn 5115 wrel 5119 wfn 5883 wf 5884 wfo 5886 |
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 |
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-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 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-fo 5894 df-fv 5896 |
This theorem is referenced by: nvo00 27616 esumnul 30110 esum0 30111 volsupnfl 33454 rnmptc 39353 |
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