Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ovmpt3rab1 Structured version   Unicode version

Theorem ovmpt3rab1 30086
 Description: The value of an operation defined by the maps-to notation with a function into a class abstraction as a result. The domain of the function and the base set of the class abstraction may depend on the operands, using implicit substitution. (Contributed by AV, 16-Jul-2018.) (Revised by AV, 16-May-2019.)
Hypotheses
Ref Expression
ovmpt3rab1.o
ovmpt3rab1.m
ovmpt3rab1.n
ovmpt3rab1.p
ovmpt3rab1.x
ovmpt3rab1.y
Assertion
Ref Expression
ovmpt3rab1
Distinct variable groups:   ,,,   ,,,   ,   ,,   ,,   ,,   ,,,,   ,,,,
Allowed substitution hints:   (,,,)   (,,,)   (,)   ()   ()   (,,,)   (,,)   (,,,)   (,)   (,)

Proof of Theorem ovmpt3rab1
StepHypRef Expression
1 ovmpt3rab1.o . . 3
21a1i 11 . 2
3 ovmpt3rab1.m . . . 4
4 ovmpt3rab1.n . . . . 5
5 ovmpt3rab1.p . . . . 5
64, 5rabeqbidv 2965 . . . 4
73, 6mpteq12dv 4367 . . 3
9 eqidd 2442 . 2
10 elex 2979 . . 3
12 elex 2979 . . 3
14 mptexg 5944 . . 3
16 nfcv 2577 . . . 4
17 nfcv 2577 . . . 4
1816, 17nfel 2585 . . 3
19 nfcv 2577 . . . 4
20 nfcv 2577 . . . 4
2119, 20nfel 2585 . . 3
22 nfcv 2577 . . . 4
23 nfcv 2577 . . . 4
2422, 23nfel 2585 . . 3
2518, 21, 24nf3an 1867 . 2
26 nfcv 2577 . . . 4
27 nfcv 2577 . . . 4
2826, 27nfel 2585 . . 3
29 nfcv 2577 . . . 4
30 nfcv 2577 . . . 4
3129, 30nfel 2585 . . 3
32 nfcv 2577 . . . 4
33 nfcv 2577 . . . 4
3432, 33nfel 2585 . . 3
3528, 31, 34nf3an 1867 . 2
36 ovmpt3rab1.x . . . 4
37 nfcv 2577 . . . 4
3836, 37nfrab 2900 . . 3
3922, 38nfmpt 4377 . 2
40 ovmpt3rab1.y . . . 4
41 nfcv 2577 . . . 4
4240, 41nfrab 2900 . . 3
4332, 42nfmpt 4377 . 2
442, 8, 9, 11, 13, 15, 25, 35, 26, 19, 39, 43ovmpt2dxf 6215 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   w3a 960   wceq 1364  wnf 1594   wcel 1761  crab 2717  cvv 2970   cmpt 4347  (class class class)co 6090   cmpt2 6092 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pr 4528 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095 This theorem is referenced by:  ovmpt3rabdm  30087  elovmpt3rab1  30088
 Copyright terms: Public domain W3C validator