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Mirrors > Home > NFE Home > Th. List > dffo3 | Unicode version |
Description: An onto mapping expressed in terms of function values. (Contributed by set.mm contributors, 29-Oct-2006.) |
Ref | Expression |
---|---|
dffo3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffo2 5274 | . 2 | |
2 | ffn 5224 | . . . . 5 | |
3 | fnrnfv 5365 | . . . . . 6 | |
4 | 3 | eqeq1d 2361 | . . . . 5 |
5 | 2, 4 | syl 15 | . . . 4 |
6 | simpr 447 | . . . . . . . . . . 11 | |
7 | ffvelrn 5416 | . . . . . . . . . . . 12 | |
8 | 7 | adantr 451 | . . . . . . . . . . 11 |
9 | 6, 8 | eqeltrd 2427 | . . . . . . . . . 10 |
10 | 9 | exp31 587 | . . . . . . . . 9 |
11 | 10 | rexlimdv 2738 | . . . . . . . 8 |
12 | 11 | biantrurd 494 | . . . . . . 7 |
13 | dfbi2 609 | . . . . . . 7 | |
14 | 12, 13 | syl6rbbr 255 | . . . . . 6 |
15 | 14 | albidv 1625 | . . . . 5 |
16 | abeq1 2460 | . . . . 5 | |
17 | df-ral 2620 | . . . . 5 | |
18 | 15, 16, 17 | 3bitr4g 279 | . . . 4 |
19 | 5, 18 | bitrd 244 | . . 3 |
20 | 19 | pm5.32i 618 | . 2 |
21 | 1, 20 | bitri 240 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 wal 1540 wceq 1642 wcel 1710 cab 2339 wral 2615 wrex 2616 crn 4774 wfn 4777 wf 4778 wfo 4780 cfv 4782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-co 4727 df-ima 4728 df-id 4768 df-cnv 4786 df-rn 4787 df-dm 4788 df-fun 4790 df-fn 4791 df-f 4792 df-fo 4794 df-fv 4796 |
This theorem is referenced by: dffo4 5424 foelrn 5426 foco2 5427 foov 5607 |
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