Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > 2ndcof | Unicode version |
Description: Composition of the second member function with another function. (Contributed by FL, 15-Oct-2012.) |
Ref | Expression |
---|---|
2ndcof |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo2nd 6109 | . . . 4 | |
2 | fofn 5397 | . . . 4 | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | ffn 5322 | . . . 4 | |
5 | dffn2 5324 | . . . 4 | |
6 | 4, 5 | sylib 121 | . . 3 |
7 | fnfco 5347 | . . 3 | |
8 | 3, 6, 7 | sylancr 411 | . 2 |
9 | rnco 5095 | . . 3 | |
10 | frn 5331 | . . . . 5 | |
11 | ssres2 4896 | . . . . 5 | |
12 | rnss 4819 | . . . . 5 | |
13 | 10, 11, 12 | 3syl 17 | . . . 4 |
14 | f2ndres 6111 | . . . . 5 | |
15 | frn 5331 | . . . . 5 | |
16 | 14, 15 | ax-mp 5 | . . . 4 |
17 | 13, 16 | sstrdi 3140 | . . 3 |
18 | 9, 17 | eqsstrid 3174 | . 2 |
19 | df-f 5177 | . 2 | |
20 | 8, 18, 19 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 cvv 2712 wss 3102 cxp 4587 crn 4590 cres 4591 ccom 4593 wfn 5168 wf 5169 wfo 5171 c2nd 6090 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-fo 5179 df-fv 5181 df-2nd 6092 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |