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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 2164 | . . . 4 | |
2 | funfvex 5497 | . . . . . 6 | |
3 | 2 | funfni 5282 | . . . . 5 |
4 | eqeq2 2174 | . . . . . . . 8 | |
5 | breq2 3980 | . . . . . . . 8 | |
6 | 4, 5 | bibi12d 234 | . . . . . . 7 |
7 | 6 | imbi2d 229 | . . . . . 6 |
8 | fneu 5286 | . . . . . . 7 | |
9 | tz6.12c 5510 | . . . . . . 7 | |
10 | 8, 9 | syl 14 | . . . . . 6 |
11 | 7, 10 | vtoclg 2781 | . . . . 5 |
12 | 3, 11 | mpcom 36 | . . . 4 |
13 | 1, 12 | mpbii 147 | . . 3 |
14 | breq2 3980 | . . 3 | |
15 | 13, 14 | syl5ibcom 154 | . 2 |
16 | fnfun 5279 | . . . 4 | |
17 | funbrfv 5519 | . . . 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 1342 weu 2013 wcel 2135 cvv 2721 class class class wbr 3976 wfun 5176 wfn 5177 cfv 5182 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 |
This theorem is referenced by: fnopfvb 5522 funbrfvb 5523 dffn5im 5526 fnsnfv 5539 fndmdif 5584 dffo4 5627 dff13 5730 isoini 5780 1stconst 6180 2ndconst 6181 pw1nct 13724 |
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