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Mirrors > Home > NFE 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). (The proof was shortened by Andrew Salmon, 22-Oct-2011.) (Contributed by set.mm contributors, 3-Aug-1994.) (Revised by set.mm contributors, 22-Oct-2011.) |
Ref | Expression |
---|---|
eqfnfv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5328 |
. . 3
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2 | 1 | ralrimivw 2699 |
. 2
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3 | pm2.27 35 |
. . . . . . . . 9
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4 | 3 | adantl 452 |
. . . . . . . 8
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5 | eqeq1 2359 |
. . . . . . . . 9
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6 | fnopfvb 5360 |
. . . . . . . . . . 11
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7 | 6 | adantlr 695 |
. . . . . . . . . 10
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8 | fnopfvb 5360 |
. . . . . . . . . . 11
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9 | 8 | adantll 694 |
. . . . . . . . . 10
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10 | 7, 9 | bibi12d 312 |
. . . . . . . . 9
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11 | 5, 10 | syl5ib 210 |
. . . . . . . 8
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12 | 4, 11 | syld 40 |
. . . . . . 7
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13 | 12 | expcom 424 |
. . . . . 6
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14 | opeldm 4911 |
. . . . . . . . . . . . 13
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15 | fndm 5183 |
. . . . . . . . . . . . . 14
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16 | 15 | eleq2d 2420 |
. . . . . . . . . . . . 13
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17 | 14, 16 | syl5ib 210 |
. . . . . . . . . . . 12
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18 | 17 | adantr 451 |
. . . . . . . . . . 11
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19 | 18 | con3d 125 |
. . . . . . . . . 10
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20 | 19 | impcom 419 |
. . . . . . . . 9
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21 | opeldm 4911 |
. . . . . . . . . . . . 13
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22 | fndm 5183 |
. . . . . . . . . . . . . 14
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23 | 22 | eleq2d 2420 |
. . . . . . . . . . . . 13
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24 | 21, 23 | syl5ib 210 |
. . . . . . . . . . . 12
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25 | 24 | adantl 452 |
. . . . . . . . . . 11
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26 | 25 | con3d 125 |
. . . . . . . . . 10
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27 | 26 | impcom 419 |
. . . . . . . . 9
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28 | 20, 27 | 2falsed 340 |
. . . . . . . 8
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29 | 28 | ex 423 |
. . . . . . 7
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30 | 29 | a1dd 42 |
. . . . . 6
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31 | 13, 30 | pm2.61i 156 |
. . . . 5
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32 | 31 | alrimdv 1633 |
. . . 4
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33 | 32 | alimdv 1621 |
. . 3
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34 | df-ral 2620 |
. . 3
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35 | eqrel 4846 |
. . 3
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36 | 33, 34, 35 | 3imtr4g 261 |
. 2
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37 | 2, 36 | impbid2 195 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-co 4727 df-ima 4728 df-id 4768 df-cnv 4786 df-rn 4787 df-dm 4788 df-fun 4790 df-fn 4791 df-fv 4796 |
This theorem is referenced by: eqfnfv2 5394 eqfnfvd 5396 eqfnfv2f 5397 fvreseq 5399 fconst2g 5453 |
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