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Mirrors > Home > ILE Home > Th. List > 1stcof | Unicode version |
Description: Composition of the first member function with another function. (Contributed by NM, 12-Oct-2007.) |
Ref | Expression |
---|---|
1stcof |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 6136 | . . . 4 | |
2 | fofn 5422 | . . . 4 | |
3 | 1, 2 | ax-mp 5 | . . 3 |
4 | ffn 5347 | . . . 4 | |
5 | dffn2 5349 | . . . 4 | |
6 | 4, 5 | sylib 121 | . . 3 |
7 | fnfco 5372 | . . 3 | |
8 | 3, 6, 7 | sylancr 412 | . 2 |
9 | rnco 5117 | . . 3 | |
10 | frn 5356 | . . . . 5 | |
11 | ssres2 4918 | . . . . 5 | |
12 | rnss 4841 | . . . . 5 | |
13 | 10, 11, 12 | 3syl 17 | . . . 4 |
14 | f1stres 6138 | . . . . 5 | |
15 | frn 5356 | . . . . 5 | |
16 | 14, 15 | ax-mp 5 | . . . 4 |
17 | 13, 16 | sstrdi 3159 | . . 3 |
18 | 9, 17 | eqsstrid 3193 | . 2 |
19 | df-f 5202 | . 2 | |
20 | 8, 18, 19 | sylanbrc 415 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 cvv 2730 wss 3121 cxp 4609 crn 4612 cres 4613 ccom 4615 wfn 5193 wf 5194 wfo 5196 c1st 6117 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fo 5204 df-fv 5206 df-1st 6119 |
This theorem is referenced by: (None) |
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