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Mirrors > Home > MPE Home > Th. List > pmvalg | Structured version Visualization version Unicode version |
Description: The value of the partial mapping operation. is the set of all partial functions that map from to . (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
pmvalg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3687 | . . 3 | |
2 | xpexg 6960 | . . . . 5 | |
3 | 2 | ancoms 469 | . . . 4 |
4 | pwexg 4850 | . . . 4 | |
5 | 3, 4 | syl 17 | . . 3 |
6 | ssexg 4804 | . . 3 | |
7 | 1, 5, 6 | sylancr 695 | . 2 |
8 | elex 3212 | . . 3 | |
9 | elex 3212 | . . 3 | |
10 | xpeq2 5129 | . . . . . . 7 | |
11 | 10 | pweqd 4163 | . . . . . 6 |
12 | rabeq 3192 | . . . . . 6 | |
13 | 11, 12 | syl 17 | . . . . 5 |
14 | xpeq1 5128 | . . . . . . 7 | |
15 | 14 | pweqd 4163 | . . . . . 6 |
16 | rabeq 3192 | . . . . . 6 | |
17 | 15, 16 | syl 17 | . . . . 5 |
18 | df-pm 7860 | . . . . 5 | |
19 | 13, 17, 18 | ovmpt2g 6795 | . . . 4 |
20 | 19 | 3expia 1267 | . . 3 |
21 | 8, 9, 20 | syl2an 494 | . 2 |
22 | 7, 21 | mpd 15 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 crab 2916 cvv 3200 wss 3574 cpw 4158 cxp 5112 wfun 5882 (class class class)co 6650 cpm 7858 |
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 ax-un 6949 |
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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 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-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pm 7860 |
This theorem is referenced by: elpmg 7873 |
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