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Mirrors > Home > ILE Home > Th. List > eqfnfv2f | Unicode version |
Description: Equality of functions is determined by their values. Special case of Exercise 4 of [TakeutiZaring] p. 28 (with domain equality omitted). This version of eqfnfv 5591 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 29-Jan-2004.) |
Ref | Expression |
---|---|
eqfnfv2f.1 | |
eqfnfv2f.2 |
Ref | Expression |
---|---|
eqfnfv2f |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqfnfv 5591 | . 2 | |
2 | eqfnfv2f.1 | . . . . 5 | |
3 | nfcv 2312 | . . . . 5 | |
4 | 2, 3 | nffv 5504 | . . . 4 |
5 | eqfnfv2f.2 | . . . . 5 | |
6 | 5, 3 | nffv 5504 | . . . 4 |
7 | 4, 6 | nfeq 2320 | . . 3 |
8 | nfv 1521 | . . 3 | |
9 | fveq2 5494 | . . . 4 | |
10 | fveq2 5494 | . . . 4 | |
11 | 9, 10 | eqeq12d 2185 | . . 3 |
12 | 7, 8, 11 | cbvral 2692 | . 2 |
13 | 1, 12 | bitrdi 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1348 wnfc 2299 wral 2448 wfn 5191 cfv 5196 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fn 5199 df-fv 5204 |
This theorem is referenced by: (None) |
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