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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 2137 | . . . 4 | |
2 | funfvex 5431 | . . . . . 6 | |
3 | 2 | funfni 5218 | . . . . 5 |
4 | eqeq2 2147 | . . . . . . . 8 | |
5 | breq2 3928 | . . . . . . . 8 | |
6 | 4, 5 | bibi12d 234 | . . . . . . 7 |
7 | 6 | imbi2d 229 | . . . . . 6 |
8 | fneu 5222 | . . . . . . 7 | |
9 | tz6.12c 5444 | . . . . . . 7 | |
10 | 8, 9 | syl 14 | . . . . . 6 |
11 | 7, 10 | vtoclg 2741 | . . . . 5 |
12 | 3, 11 | mpcom 36 | . . . 4 |
13 | 1, 12 | mpbii 147 | . . 3 |
14 | breq2 3928 | . . 3 | |
15 | 13, 14 | syl5ibcom 154 | . 2 |
16 | fnfun 5215 | . . . 4 | |
17 | funbrfv 5453 | . . . 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 1997 cvv 2681 class class class wbr 3924 wfun 5112 wfn 5113 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-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fn 5121 df-fv 5126 |
This theorem is referenced by: fnopfvb 5456 funbrfvb 5457 dffn5im 5460 fnsnfv 5473 fndmdif 5518 dffo4 5561 dff13 5662 isoini 5712 1stconst 6111 2ndconst 6112 |
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