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Mirrors > Home > ILE Home > Th. List > fnbrfvb | Unicode version |
Description: Equivalence of function value and binary relation. (Contributed by NM, 19-Apr-2004.) (Revised by Mario Carneiro, 28-Apr-2015.) |
Ref | Expression |
---|---|
fnbrfvb |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . . 4 | |
2 | funfvex 5438 | . . . . . 6 | |
3 | 2 | funfni 5223 | . . . . 5 |
4 | eqeq2 2149 | . . . . . . . 8 | |
5 | breq2 3933 | . . . . . . . 8 | |
6 | 4, 5 | bibi12d 234 | . . . . . . 7 |
7 | 6 | imbi2d 229 | . . . . . 6 |
8 | fneu 5227 | . . . . . . 7 | |
9 | tz6.12c 5451 | . . . . . . 7 | |
10 | 8, 9 | syl 14 | . . . . . 6 |
11 | 7, 10 | vtoclg 2746 | . . . . 5 |
12 | 3, 11 | mpcom 36 | . . . 4 |
13 | 1, 12 | mpbii 147 | . . 3 |
14 | breq2 3933 | . . 3 | |
15 | 13, 14 | syl5ibcom 154 | . 2 |
16 | fnfun 5220 | . . . 4 | |
17 | funbrfv 5460 | . . . 4 | |
18 | 16, 17 | syl 14 | . . 3 |
19 | 18 | adantr 274 | . 2 |
20 | 15, 19 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 weu 1999 cvv 2686 class class class wbr 3929 wfun 5117 wfn 5118 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-iota 5088 df-fun 5125 df-fn 5126 df-fv 5131 |
This theorem is referenced by: fnopfvb 5463 funbrfvb 5464 dffn5im 5467 fnsnfv 5480 fndmdif 5525 dffo4 5568 dff13 5669 isoini 5719 1stconst 6118 2ndconst 6119 |
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