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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 5327 | . . 3 | |
2 | 1 | ralrimivw 2698 | . 2 |
3 | pm2.27 35 | . . . . . . . . 9 | |
4 | 3 | adantl 452 | . . . . . . . 8 |
5 | eqeq1 2359 | . . . . . . . . 9 | |
6 | fnopfvb 5359 | . . . . . . . . . . 11 | |
7 | 6 | adantlr 695 | . . . . . . . . . 10 |
8 | fnopfvb 5359 | . . . . . . . . . . 11 | |
9 | 8 | adantll 694 | . . . . . . . . . 10 |
10 | 7, 9 | bibi12d 312 | . . . . . . . . 9 |
11 | 5, 10 | syl5ib 210 | . . . . . . . 8 |
12 | 4, 11 | syld 40 | . . . . . . 7 |
13 | 12 | expcom 424 | . . . . . 6 |
14 | opeldm 4910 | . . . . . . . . . . . . 13 | |
15 | fndm 5182 | . . . . . . . . . . . . . 14 | |
16 | 15 | eleq2d 2420 | . . . . . . . . . . . . 13 |
17 | 14, 16 | syl5ib 210 | . . . . . . . . . . . 12 |
18 | 17 | adantr 451 | . . . . . . . . . . 11 |
19 | 18 | con3d 125 | . . . . . . . . . 10 |
20 | 19 | impcom 419 | . . . . . . . . 9 |
21 | opeldm 4910 | . . . . . . . . . . . . 13 | |
22 | fndm 5182 | . . . . . . . . . . . . . 14 | |
23 | 22 | eleq2d 2420 | . . . . . . . . . . . . 13 |
24 | 21, 23 | syl5ib 210 | . . . . . . . . . . . 12 |
25 | 24 | adantl 452 | . . . . . . . . . . 11 |
26 | 25 | con3d 125 | . . . . . . . . . 10 |
27 | 26 | impcom 419 | . . . . . . . . 9 |
28 | 20, 27 | 2falsed 340 | . . . . . . . 8 |
29 | 28 | ex 423 | . . . . . . 7 |
30 | 29 | a1dd 42 | . . . . . 6 |
31 | 13, 30 | pm2.61i 156 | . . . . 5 |
32 | 31 | alrimdv 1633 | . . . 4 |
33 | 32 | alimdv 1621 | . . 3 |
34 | df-ral 2619 | . . 3 | |
35 | eqrel 4845 | . . 3 | |
36 | 33, 34, 35 | 3imtr4g 261 | . 2 |
37 | 2, 36 | impbid2 195 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wb 176 wa 358 wal 1540 wceq 1642 wcel 1710 wral 2614 cop 4561 cdm 4772 wfn 4776 cfv 4781 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
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 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-fv 4795 |
This theorem is referenced by: eqfnfv2 5393 eqfnfvd 5395 eqfnfv2f 5396 fvreseq 5398 fconst2g 5452 |
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