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Theorem elfvmptrab1 5985
 Description: Implications for the value of a function defined by the maps-to notation with a class abstraction as a result having an element. Here, the base set of the class abstraction depends on the argument of the function. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elfvmptrab1.f
elfvmptrab1.v
Assertion
Ref Expression
elfvmptrab1
Distinct variable groups:   ,,   ,   ,,   ,   ,
Allowed substitution hints:   (,,)   (,,)   ()   (,)   ()   (,)

Proof of Theorem elfvmptrab1
StepHypRef Expression
1 ne0i 3728 . . 3
2 ndmfv 5903 . . . 4
32necon1ai 2670 . . 3
4 elfvmptrab1.f . . . . . . . 8
54dmmptss 5338 . . . . . . 7
65sseli 3414 . . . . . 6
7 elfvmptrab1.v . . . . . . 7
8 rabexg 4549 . . . . . . 7
96, 7, 83syl 18 . . . . . 6
10 nfcv 2612 . . . . . . 7
11 nfsbc1v 3275 . . . . . . . 8
12 nfcv 2612 . . . . . . . . 9
1310, 12nfcsb 3367 . . . . . . . 8
1411, 13nfrab 2958 . . . . . . 7
15 csbeq1 3352 . . . . . . . 8
16 sbceq1a 3266 . . . . . . . 8
1715, 16rabeqbidv 3026 . . . . . . 7
1810, 14, 17, 4fvmptf 5981 . . . . . 6
196, 9, 18syl2anc 673 . . . . 5
2019eleq2d 2534 . . . 4
21 elrabi 3181 . . . . . 6
226, 21anim12i 576 . . . . 5
2322ex 441 . . . 4
2420, 23sylbid 223 . . 3
251, 3, 243syl 18 . 2
2625pm2.43i 48 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452   wcel 1904   wne 2641  crab 2760  cvv 3031  wsbc 3255  csb 3349  c0 3722   cmpt 4454   cdm 4839  cfv 5589 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597 This theorem is referenced by:  elfvmptrab  5986
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