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Mirrors > Home > ILE Home > Th. List > ffnfv | Unicode version |
Description: A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.) |
Ref | Expression |
---|---|
ffnfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5267 | . . 3 | |
2 | ffvelrn 5546 | . . . 4 | |
3 | 2 | ralrimiva 2503 | . . 3 |
4 | 1, 3 | jca 304 | . 2 |
5 | simpl 108 | . . 3 | |
6 | fvelrnb 5462 | . . . . . 6 | |
7 | 6 | biimpd 143 | . . . . 5 |
8 | nfra1 2464 | . . . . . 6 | |
9 | nfv 1508 | . . . . . 6 | |
10 | rsp 2478 | . . . . . . 7 | |
11 | eleq1 2200 | . . . . . . . 8 | |
12 | 11 | biimpcd 158 | . . . . . . 7 |
13 | 10, 12 | syl6 33 | . . . . . 6 |
14 | 8, 9, 13 | rexlimd 2544 | . . . . 5 |
15 | 7, 14 | sylan9 406 | . . . 4 |
16 | 15 | ssrdv 3098 | . . 3 |
17 | df-f 5122 | . . 3 | |
18 | 5, 16, 17 | sylanbrc 413 | . 2 |
19 | 4, 18 | impbii 125 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 wrex 2415 wss 3066 crn 4535 wfn 5113 wf 5114 cfv 5118 |
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-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 |
This theorem is referenced by: ffnfvf 5572 fnfvrnss 5573 fmpt2d 5575 ffnov 5868 elixpconst 6593 elixpsn 6622 ctssdccl 6989 cnref1o 9433 shftf 10595 eff2 11375 reeff1 11396 dvfre 12832 isomninnlem 13214 |
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