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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 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffn5im 5561 |
. . 3
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2 | dffn5im 5561 |
. . 3
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3 | 1, 2 | eqeqan12d 2193 |
. 2
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4 | funfvex 5532 |
. . . . . 6
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5 | 4 | funfni 5316 |
. . . . 5
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6 | 5 | ralrimiva 2550 |
. . . 4
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7 | mpteqb 5606 |
. . . 4
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8 | 6, 7 | syl 14 |
. . 3
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9 | 8 | adantr 276 |
. 2
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10 | 3, 9 | 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 4121 ax-pow 4174 ax-pr 4209 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 |
This theorem is referenced by: eqfnfv2 5614 eqfnfvd 5616 eqfnfv2f 5617 fvreseq 5619 fnmptfvd 5620 fneqeql 5624 fconst2g 5731 cocan1 5787 cocan2 5788 tfri3 6367 updjud 7080 nninfwlporlemd 7169 ser0f 10512 prodf1f 11546 1arithlem4 12358 1arith 12359 isgrpinv 12880 cnmpt11 13676 cnmpt21 13684 |
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