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Theorem mpt2curryvald 7018
 Description: The value of a curried operation given in maps-to notation is a function over the second argument of the original operation. (Contributed by AV, 27-Oct-2019.)
Hypotheses
Ref Expression
mpt2curryd.f
mpt2curryd.c
mpt2curryd.n
mpt2curryvald.y
mpt2curryvald.a
Assertion
Ref Expression
mpt2curryvald
Distinct variable groups:   ,,   ,,   ,,   ,,   ,,   ,,

Proof of Theorem mpt2curryvald
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 mpt2curryd.f . . . 4
2 mpt2curryd.c . . . 4
3 mpt2curryd.n . . . 4
41, 2, 3mpt2curryd 7017 . . 3
5 nfcv 2619 . . . 4
6 nfcv 2619 . . . . 5
7 nfcsb1v 3450 . . . . 5
86, 7nfmpt 4540 . . . 4
9 csbeq1a 3443 . . . . 5
109mpteq2dv 4539 . . . 4
115, 8, 10cbvmpt 4542 . . 3
124, 11syl6eq 2514 . 2
13 csbeq1 3437 . . . 4
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  =wceq 1395  e.wcel 1818  =/=wne 2652  A.wral 2807   cvv 3109  [_csb 3434   c0 3784  e.cmpt 4510  cfv 5593  e.cmpt2 6298  curry`ccur 7013 This theorem is referenced by:  fvmpt2curryd  7019  pmatcollpw3lem  19284 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6800  df-2nd 6801  df-cur 7015