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Theorem fsplit 6454
 Description: A function that can be used to feed a common value to both operands of an operation. Use as the second argument of a composition with the function of fpar 6453 in order to build compound functions such as . (Contributed by NM, 17-Sep-2007.)
Assertion
Ref Expression
fsplit

Proof of Theorem fsplit
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2961 . . . . 5
2 vex 2961 . . . . 5
31, 2brcnv 5058 . . . 4
41brres 5155 . . . . 5
5 19.42v 1929 . . . . . . 7
6 vex 2961 . . . . . . . . . . 11
76, 6op1std 6360 . . . . . . . . . 10
87eqeq1d 2446 . . . . . . . . 9
98pm5.32ri 621 . . . . . . . 8
109exbii 1593 . . . . . . 7
11 fo1st 6369 . . . . . . . . . 10
12 fofn 5658 . . . . . . . . . 10
1311, 12ax-mp 5 . . . . . . . . 9
14 fnbrfvb 5770 . . . . . . . . 9
1513, 2, 14mp2an 655 . . . . . . . 8
16 dfid2 4503 . . . . . . . . . 10
1716eleq2i 2502 . . . . . . . . 9
18 nfe1 1748 . . . . . . . . . . 11
191819.9 1798 . . . . . . . . . 10
20 elopab 4465 . . . . . . . . . 10
21 equid 1689 . . . . . . . . . . . 12
2221biantru 493 . . . . . . . . . . 11
2322exbii 1593 . . . . . . . . . 10
2419, 20, 233bitr4i 270 . . . . . . . . 9
2517, 24bitr2i 243 . . . . . . . 8
2615, 25anbi12i 680 . . . . . . 7
275, 10, 263bitr3ri 269 . . . . . 6
28 id 21 . . . . . . . . 9
2928, 28opeq12d 3994 . . . . . . . 8
3029eqeq2d 2449 . . . . . . 7
311, 30ceqsexv 2993 . . . . . 6
3227, 31bitri 242 . . . . 5
334, 32bitri 242 . . . 4
343, 33bitri 242 . . 3
3534opabbii 4275 . 2
36 relcnv 5245 . . 3
37 dfrel4v 5325 . . 3
3836, 37mpbi 201 . 2
39 mptv 4304 . 2
4035, 38, 393eqtr4i 2468 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360  wex 1551   wceq 1653   wcel 1726  cvv 2958  cop 3819   class class class wbr 4215  copab 4268   cmpt 4269   cid 4496  ccnv 4880   cres 4883   wrel 4886   wfn 5452  wfo 5455  cfv 5457  c1st 6350 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fo 5463  df-fv 5465  df-1st 6352
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