Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dmcoss | Structured version Visualization version Unicode version |
Description: Domain of a composition. Theorem 21 of [Suppes] p. 63. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmcoss |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2027 | . . . 4 | |
2 | exsimpl 1795 | . . . . 5 | |
3 | vex 3203 | . . . . . 6 | |
4 | vex 3203 | . . . . . 6 | |
5 | 3, 4 | opelco 5293 | . . . . 5 |
6 | breq2 4657 | . . . . . 6 | |
7 | 6 | cbvexv 2275 | . . . . 5 |
8 | 2, 5, 7 | 3imtr4i 281 | . . . 4 |
9 | 1, 8 | exlimi 2086 | . . 3 |
10 | 3 | eldm2 5322 | . . 3 |
11 | 3 | eldm 5321 | . . 3 |
12 | 9, 10, 11 | 3imtr4i 281 | . 2 |
13 | 12 | ssriv 3607 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 384 wex 1704 wcel 1990 wss 3574 cop 4183 class class class wbr 4653 cdm 5114 ccom 5118 |
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-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-co 5123 df-dm 5124 |
This theorem is referenced by: rncoss 5386 dmcosseq 5387 cossxp 5658 fvco4i 6276 cofunexg 7130 fin23lem30 9164 wunco 9555 relexpnndm 13781 mvdco 17865 f1omvdconj 17866 znleval 19903 ofco2 20257 tngtopn 22454 xppreima 29449 relexp0a 38008 dmtrclfvRP 38022 |
Copyright terms: Public domain | W3C validator |