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Mirrors > Home > ILE Home > Th. List > eqfnfv2 | Unicode version |
Description: Equality of functions is determined by their values. Exercise 4 of [TakeutiZaring] p. 28. (Contributed by NM, 3-Aug-1994.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
eqfnfv2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 4823 |
. . . 4
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2 | fndm 5311 |
. . . . 5
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3 | fndm 5311 |
. . . . 5
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4 | 2, 3 | eqeqan12d 2193 |
. . . 4
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5 | 1, 4 | imbitrid 154 |
. . 3
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6 | 5 | pm4.71rd 394 |
. 2
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7 | fneq2 5301 |
. . . . . 6
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8 | 7 | biimparc 299 |
. . . . 5
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9 | eqfnfv 5609 |
. . . . 5
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10 | 8, 9 | sylan2 286 |
. . . 4
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11 | 10 | anassrs 400 |
. . 3
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12 | 11 | pm5.32da 452 |
. 2
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13 | 6, 12 | bitrd 188 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fn 5215 df-fv 5220 |
This theorem is referenced by: eqfnfv3 5611 eqfunfv 5614 eqfnov 5975 2ffzeq 10124 |
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