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Mirrors > Home > ILE Home > Th. List > dff3im | Unicode version |
Description: Property of a mapping. (Contributed by Jim Kingdon, 4-Jan-2019.) |
Ref | Expression |
---|---|
dff3im |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fssxp 5290 | . 2 | |
2 | ffun 5275 | . . . . . . . 8 | |
3 | 2 | adantr 274 | . . . . . . 7 |
4 | fdm 5278 | . . . . . . . . 9 | |
5 | 4 | eleq2d 2209 | . . . . . . . 8 |
6 | 5 | biimpar 295 | . . . . . . 7 |
7 | funfvop 5532 | . . . . . . 7 | |
8 | 3, 6, 7 | syl2anc 408 | . . . . . 6 |
9 | df-br 3930 | . . . . . 6 | |
10 | 8, 9 | sylibr 133 | . . . . 5 |
11 | funfvex 5438 | . . . . . . 7 | |
12 | breq2 3933 | . . . . . . . 8 | |
13 | 12 | spcegv 2774 | . . . . . . 7 |
14 | 11, 13 | syl 14 | . . . . . 6 |
15 | 3, 6, 14 | syl2anc 408 | . . . . 5 |
16 | 10, 15 | mpd 13 | . . . 4 |
17 | funmo 5138 | . . . . . 6 | |
18 | 2, 17 | syl 14 | . . . . 5 |
19 | 18 | adantr 274 | . . . 4 |
20 | eu5 2046 | . . . 4 | |
21 | 16, 19, 20 | sylanbrc 413 | . . 3 |
22 | 21 | ralrimiva 2505 | . 2 |
23 | 1, 22 | jca 304 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wex 1468 wcel 1480 weu 1999 wmo 2000 wral 2416 cvv 2686 wss 3071 cop 3530 class class class wbr 3929 cxp 4537 cdm 4539 wfun 5117 wf 5119 cfv 5123 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 |
This theorem is referenced by: dff4im 5566 |
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