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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 4887 | . . 3 | |
2 | dfdm4 4726 | . . . 4 | |
3 | fdm 5273 | . . . . 5 | |
4 | ssid 3112 | . . . . 5 | |
5 | 3, 4 | eqsstrdi 3144 | . . . 4 |
6 | 2, 5 | eqsstrrid 3139 | . . 3 |
7 | 1, 6 | sstrid 3103 | . 2 |
8 | imassrn 4887 | . . . 4 | |
9 | frn 5276 | . . . 4 | |
10 | 8, 9 | sstrid 3103 | . . 3 |
11 | ffun 5270 | . . . 4 | |
12 | 4, 3 | sseqtrrid 3143 | . . . 4 |
13 | funimass3 5529 | . . . 4 | |
14 | 11, 12, 13 | syl2anc 408 | . . 3 |
15 | 10, 14 | mpbid 146 | . 2 |
16 | 7, 15 | eqssd 3109 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 wceq 1331 wss 3066 ccnv 4533 cdm 4534 crn 4535 cima 4537 wfun 5112 wf 5114 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 |
This theorem is referenced by: fmpt 5563 nn0supp 9022 cnclima 12381 |
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