Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fpr | Structured version Visualization version Unicode version |
Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
fpr.1 | |
fpr.2 | |
fpr.3 | |
fpr.4 |
Ref | Expression |
---|---|
fpr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fpr.1 | . . . . . 6 | |
2 | fpr.2 | . . . . . 6 | |
3 | fpr.3 | . . . . . 6 | |
4 | fpr.4 | . . . . . 6 | |
5 | 1, 2, 3, 4 | funpr 5944 | . . . . 5 |
6 | 3, 4 | dmprop 5610 | . . . . 5 |
7 | 5, 6 | jctir 561 | . . . 4 |
8 | df-fn 5891 | . . . 4 | |
9 | 7, 8 | sylibr 224 | . . 3 |
10 | df-pr 4180 | . . . . . 6 | |
11 | 10 | rneqi 5352 | . . . . 5 |
12 | rnun 5541 | . . . . 5 | |
13 | 1 | rnsnop 5616 | . . . . . . 7 |
14 | 2 | rnsnop 5616 | . . . . . . 7 |
15 | 13, 14 | uneq12i 3765 | . . . . . 6 |
16 | df-pr 4180 | . . . . . 6 | |
17 | 15, 16 | eqtr4i 2647 | . . . . 5 |
18 | 11, 12, 17 | 3eqtri 2648 | . . . 4 |
19 | 18 | eqimssi 3659 | . . 3 |
20 | 9, 19 | jctir 561 | . 2 |
21 | df-f 5892 | . 2 | |
22 | 20, 21 | sylibr 224 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wne 2794 cvv 3200 cun 3572 wss 3574 csn 4177 cpr 4179 cop 4183 cdm 5114 crn 5115 wfun 5882 wfn 5883 wf 5884 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 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-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-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-fun 5890 df-fn 5891 df-f 5892 |
This theorem is referenced by: fprg 6422 1sdom 8163 axlowdimlem4 25825 coinfliprv 30544 fprb 31669 poimirlem22 33431 nnsum3primes4 41676 nnsum3primesgbe 41680 |
Copyright terms: Public domain | W3C validator |