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Mirrors > Home > NFE Home > Th. List > resdif | Unicode version |
Description: The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.) |
Ref | Expression |
---|---|
resdif |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun 5270 | . . . . . 6 | |
2 | difss 3393 | . . . . . . 7 | |
3 | fof 5269 | . . . . . . . 8 | |
4 | fdm 5226 | . . . . . . . 8 | |
5 | 3, 4 | syl 15 | . . . . . . 7 |
6 | 2, 5 | syl5sseqr 3320 | . . . . . 6 |
7 | fores 5278 | . . . . . 6 | |
8 | 1, 6, 7 | syl2anc 642 | . . . . 5 |
9 | resabs1 4992 | . . . . . . . 8 | |
10 | 2, 9 | ax-mp 5 | . . . . . . 7 |
11 | foeq1 5265 | . . . . . . 7 | |
12 | 10, 11 | ax-mp 5 | . . . . . 6 |
13 | 10 | rneqi 4957 | . . . . . . . 8 |
14 | dfima3 4951 | . . . . . . . 8 | |
15 | dfima3 4951 | . . . . . . . 8 | |
16 | 13, 14, 15 | 3eqtr4i 2383 | . . . . . . 7 |
17 | foeq3 5267 | . . . . . . 7 | |
18 | 16, 17 | ax-mp 5 | . . . . . 6 |
19 | 12, 18 | bitri 240 | . . . . 5 |
20 | 8, 19 | sylib 188 | . . . 4 |
21 | funres11 5164 | . . . 4 | |
22 | dff1o3 5292 | . . . . 5 | |
23 | 22 | biimpri 197 | . . . 4 |
24 | 20, 21, 23 | syl2anr 464 | . . 3 |
25 | 24 | 3adant3 975 | . 2 |
26 | dfima3 4951 | . . . . . . 7 | |
27 | forn 5272 | . . . . . . 7 | |
28 | 26, 27 | syl5eq 2397 | . . . . . 6 |
29 | dfima3 4951 | . . . . . . 7 | |
30 | forn 5272 | . . . . . . 7 | |
31 | 29, 30 | syl5eq 2397 | . . . . . 6 |
32 | 28, 31 | anim12i 549 | . . . . 5 |
33 | imadif 5171 | . . . . . 6 | |
34 | difeq12 3380 | . . . . . 6 | |
35 | 33, 34 | sylan9eq 2405 | . . . . 5 |
36 | 32, 35 | sylan2 460 | . . . 4 |
37 | 36 | 3impb 1147 | . . 3 |
38 | f1oeq3 5283 | . . 3 | |
39 | 37, 38 | syl 15 | . 2 |
40 | 25, 39 | mpbid 201 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 176 wa 358 w3a 934 wceq 1642 cdif 3206 wss 3257 cima 4722 ccnv 4771 cdm 4772 crn 4773 cres 4774 wfun 4775 wf 4777 wfo 4779 wf1o 4780 |
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-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 |
This theorem is referenced by: resin 5307 |
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