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Theorem fnresin 28231
 Description: Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)
Assertion
Ref Expression
fnresin

Proof of Theorem fnresin
StepHypRef Expression
1 fnresin1 5708 . 2
2 resindi 5139 . . . 4
3 fnresdm 5703 . . . . . 6
43ineq1d 3663 . . . . 5
5 incom 3655 . . . . . 6
6 resss 5147 . . . . . . 7
7 df-ss 3450 . . . . . . 7
86, 7mpbi 211 . . . . . 6
95, 8eqtr3i 2453 . . . . 5
104, 9syl6eq 2479 . . . 4
112, 10syl5eq 2475 . . 3
1211fneq1d 5684 . 2
131, 12mpbid 213 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wceq 1437   cin 3435   wss 3436   cres 4855   wfn 5596 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-res 4865  df-fun 5603  df-fn 5604 This theorem is referenced by:  signstres  29473
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