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Theorem elovmpt2rab1 6535
 Description: Implications for the value of an operation, 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 first operand. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Hypotheses
Ref Expression
elovmpt2rab1.o
elovmpt2rab1.v
Assertion
Ref Expression
elovmpt2rab1
Distinct variable groups:   ,,,   ,,,   ,,,   ,   ,
Allowed substitution hints:   (,,,)   ()   (,,,)   ()   ()   (,,)

Proof of Theorem elovmpt2rab1
StepHypRef Expression
1 elovmpt2rab1.o . . 3
21elmpt2cl 6530 . 2
31a1i 11 . . . . 5
4 csbeq1 3352 . . . . . . 7
54ad2antrl 742 . . . . . 6
6 sbceq1a 3266 . . . . . . . 8
7 sbceq1a 3266 . . . . . . . 8
86, 7sylan9bbr 715 . . . . . . 7
98adantl 473 . . . . . 6
105, 9rabeqbidv 3026 . . . . 5
11 eqidd 2472 . . . . 5
12 simpl 464 . . . . 5
13 simpr 468 . . . . 5
14 elovmpt2rab1.v . . . . . 6
15 rabexg 4549 . . . . . 6
1614, 15syl 17 . . . . 5
17 nfcv 2612 . . . . . . 7
1817nfel1 2626 . . . . . 6
19 nfcv 2612 . . . . . . 7
2019nfel1 2626 . . . . . 6
2118, 20nfan 2031 . . . . 5
22 nfcv 2612 . . . . . . 7
2322nfel1 2626 . . . . . 6
24 nfcv 2612 . . . . . . 7
2524nfel1 2626 . . . . . 6
2623, 25nfan 2031 . . . . 5
27 nfsbc1v 3275 . . . . . 6
28 nfcv 2612 . . . . . . 7
2917, 28nfcsb 3367 . . . . . 6
3027, 29nfrab 2958 . . . . 5
31 nfsbc1v 3275 . . . . . . 7
3222, 31nfsbc 3277 . . . . . 6
33 nfcv 2612 . . . . . . 7
3422, 33nfcsb 3367 . . . . . 6
3532, 34nfrab 2958 . . . . 5
363, 10, 11, 12, 13, 16, 21, 26, 22, 19, 30, 35ovmpt2dxf 6441 . . . 4
3736eleq2d 2534 . . 3
38 elrabi 3181 . . . . 5
39 df-3an 1009 . . . . . 6
4039simplbi2com 639 . . . . 5
4138, 40syl 17 . . . 4
4241com12 31 . . 3
4337, 42sylbid 223 . 2
442, 43mpcom 36 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   w3a 1007   wceq 1452   wcel 1904  crab 2760  cvv 3031  wsbc 3255  csb 3349  (class class class)co 6308   cmpt2 6310 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-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313 This theorem is referenced by:  elovmpt2wrd  12760
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