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Mirrors > Home > NFE Home > Th. List > mptpreima | Unicode version |
Description: The preimage of a function in maps-to notation. (Contributed by Stefan O'Rear, 25-Jan-2015.) |
Ref | Expression |
---|---|
dmmpt2.1 |
Ref | Expression |
---|---|
mptpreima |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmmpt2.1 | . . . . . 6 | |
2 | df-mpt 5652 | . . . . . 6 | |
3 | 1, 2 | eqtri 2373 | . . . . 5 |
4 | 3 | cnveqi 4887 | . . . 4 |
5 | cnvopab 5030 | . . . 4 | |
6 | 4, 5 | eqtri 2373 | . . 3 |
7 | 6 | imaeq1i 4939 | . 2 |
8 | dfima3 4951 | . . 3 | |
9 | resopab 4999 | . . . . 5 | |
10 | 9 | rneqi 4957 | . . . 4 |
11 | ancom 437 | . . . . . . . . 9 | |
12 | anass 630 | . . . . . . . . 9 | |
13 | 11, 12 | bitri 240 | . . . . . . . 8 |
14 | 13 | exbii 1582 | . . . . . . 7 |
15 | 19.42v 1905 | . . . . . . . 8 | |
16 | df-clel 2349 | . . . . . . . . . 10 | |
17 | 16 | bicomi 193 | . . . . . . . . 9 |
18 | 17 | anbi2i 675 | . . . . . . . 8 |
19 | 15, 18 | bitri 240 | . . . . . . 7 |
20 | 14, 19 | bitri 240 | . . . . . 6 |
21 | 20 | abbii 2465 | . . . . 5 |
22 | rnopab 4967 | . . . . 5 | |
23 | df-rab 2623 | . . . . 5 | |
24 | 21, 22, 23 | 3eqtr4i 2383 | . . . 4 |
25 | 10, 24 | eqtri 2373 | . . 3 |
26 | 8, 25 | eqtri 2373 | . 2 |
27 | 7, 26 | eqtri 2373 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 358 wex 1541 wceq 1642 wcel 1710 cab 2339 crab 2618 copab 4622 cima 4722 ccnv 4771 crn 4773 cres 4774 cmpt 5651 |
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-ima 4727 df-xp 4784 df-cnv 4785 df-rn 4786 df-res 4788 df-mpt 5652 |
This theorem is referenced by: fmpt 5692 |
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