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Mirrors > Home > MPE Home > Th. List > funfvima | Structured version Visualization version Unicode version |
Description: A function's value in a preimage belongs to the image. (Contributed by NM, 23-Sep-2003.) |
Ref | Expression |
---|---|
funfvima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmres 5419 | . . . . . . 7 | |
2 | 1 | elin2 3801 | . . . . . 6 |
3 | funres 5929 | . . . . . . . . 9 | |
4 | fvelrn 6352 | . . . . . . . . 9 | |
5 | 3, 4 | sylan 488 | . . . . . . . 8 |
6 | fvres 6207 | . . . . . . . . . 10 | |
7 | 6 | eleq1d 2686 | . . . . . . . . 9 |
8 | df-ima 5127 | . . . . . . . . . 10 | |
9 | 8 | eleq2i 2693 | . . . . . . . . 9 |
10 | 7, 9 | syl6rbbr 279 | . . . . . . . 8 |
11 | 5, 10 | syl5ibrcom 237 | . . . . . . 7 |
12 | 11 | ex 450 | . . . . . 6 |
13 | 2, 12 | syl5bir 233 | . . . . 5 |
14 | 13 | expd 452 | . . . 4 |
15 | 14 | com12 32 | . . 3 |
16 | 15 | impd 447 | . 2 |
17 | 16 | pm2.43b 55 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wcel 1990 cdm 5114 crn 5115 cres 5116 cima 5117 wfun 5882 cfv 5888 |
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-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-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-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-fv 5896 |
This theorem is referenced by: funfvima2 6493 elovimad 6693 tz7.48-2 7537 tz9.12lem3 8652 lindff1 20159 txcnp 21423 c1liplem1 23759 pthdivtx 26625 htthlem 27774 tpr2rico 29958 brsiga 30246 erdszelem8 31180 relowlpssretop 33212 limsuppnfdlem 39933 limsupresxr 39998 liminfresxr 39999 liminfvalxr 40015 |
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