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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 5273 | . 2 | |
2 | ffn 5223 | . . . . 5 | |
3 | fnrnfv 5364 | . . . . . 6 | |
4 | 3 | eqeq1d 2361 | . . . . 5 |
5 | 2, 4 | syl 15 | . . . 4 |
6 | simpr 447 | . . . . . . . . . . 11 | |
7 | ffvelrn 5415 | . . . . . . . . . . . 12 | |
8 | 7 | adantr 451 | . . . . . . . . . . 11 |
9 | 6, 8 | eqeltrd 2427 | . . . . . . . . . 10 |
10 | 9 | exp31 587 | . . . . . . . . 9 |
11 | 10 | rexlimdv 2737 | . . . . . . . 8 |
12 | 11 | biantrurd 494 | . . . . . . 7 |
13 | dfbi2 609 | . . . . . . 7 | |
14 | 12, 13 | syl6rbbr 255 | . . . . . 6 |
15 | 14 | albidv 1625 | . . . . 5 |
16 | abeq1 2459 | . . . . 5 | |
17 | df-ral 2619 | . . . . 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 2614 wrex 2615 crn 4773 wfn 4776 wf 4777 wfo 4779 cfv 4781 |
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 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
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 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 df-fo 4793 df-fv 4795 |
This theorem is referenced by: dffo4 5423 foelrn 5425 foco2 5426 foov 5606 |
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