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Mirrors > Home > ILE Home > Th. List > fimacnv | Unicode version |
Description: The preimage of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.) |
Ref | Expression |
---|---|
fimacnv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imassrn 4862 | . . 3 | |
2 | dfdm4 4701 | . . . 4 | |
3 | fdm 5248 | . . . . 5 | |
4 | ssid 3087 | . . . . 5 | |
5 | 3, 4 | eqsstrdi 3119 | . . . 4 |
6 | 2, 5 | eqsstrrid 3114 | . . 3 |
7 | 1, 6 | sstrid 3078 | . 2 |
8 | imassrn 4862 | . . . 4 | |
9 | frn 5251 | . . . 4 | |
10 | 8, 9 | sstrid 3078 | . . 3 |
11 | ffun 5245 | . . . 4 | |
12 | 4, 3 | sseqtrrid 3118 | . . . 4 |
13 | funimass3 5504 | . . . 4 | |
14 | 11, 12, 13 | syl2anc 408 | . . 3 |
15 | 10, 14 | mpbid 146 | . 2 |
16 | 7, 15 | eqssd 3084 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1316 wss 3041 ccnv 4508 cdm 4509 crn 4510 cima 4512 wfun 5087 wf 5089 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 |
This theorem is referenced by: fmpt 5538 nn0supp 8997 cnclima 12319 |
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