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Mirrors > Home > ILE Home > Th. List > eqfnfv | Unicode version |
Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
eqfnfv |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn5im 5467 | . . 3 | |
2 | dffn5im 5467 | . . 3 | |
3 | 1, 2 | eqeqan12d 2155 | . 2 |
4 | funfvex 5438 | . . . . . 6 | |
5 | 4 | funfni 5223 | . . . . 5 |
6 | 5 | ralrimiva 2505 | . . . 4 |
7 | mpteqb 5511 | . . . 4 | |
8 | 6, 7 | syl 14 | . . 3 |
9 | 8 | adantr 274 | . 2 |
10 | 3, 9 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 cvv 2686 cmpt 3989 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-csb 3004 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-mpt 3991 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: eqfnfv2 5519 eqfnfvd 5521 eqfnfv2f 5522 fvreseq 5524 fneqeql 5528 fconst2g 5635 cocan1 5688 cocan2 5689 tfri3 6264 updjud 6967 ser0f 10288 prodf1f 11312 cnmpt11 12452 cnmpt21 12460 |
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