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Theorem offval2f 24033
 Description: The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)
Hypotheses
Ref Expression
offval2f.0
offval2f.a
offval2f.1
offval2f.2
offval2f.3
offval2f.4
offval2f.5
Assertion
Ref Expression
offval2f
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()   ()   ()   ()   ()   ()

Proof of Theorem offval2f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 offval2f.0 . . . . . 6
2 offval2f.2 . . . . . . 7
32ex 424 . . . . . 6
41, 3ralrimi 2747 . . . . 5
5 offval2f.a . . . . . 6
65fnmptf 24027 . . . . 5
74, 6syl 16 . . . 4
8 offval2f.4 . . . . 5
98fneq1d 5495 . . . 4
107, 9mpbird 224 . . 3
11 offval2f.3 . . . . . . 7
1211ex 424 . . . . . 6
131, 12ralrimi 2747 . . . . 5
145fnmptf 24027 . . . . 5
1513, 14syl 16 . . . 4
16 offval2f.5 . . . . 5
1716fneq1d 5495 . . . 4
1815, 17mpbird 224 . . 3
19 offval2f.1 . . 3
20 inidm 3510 . . 3
218adantr 452 . . . 4
2221fveq1d 5689 . . 3
2316adantr 452 . . . 4
2423fveq1d 5689 . . 3
2510, 18, 19, 19, 20, 22, 24offval 6271 . 2
26 nfcv 2540 . . . 4
27 nffvmpt1 5695 . . . . 5
28 nfcv 2540 . . . . 5
29 nffvmpt1 5695 . . . . 5
3027, 28, 29nfov 6063 . . . 4
31 nfcv 2540 . . . 4
32 fveq2 5687 . . . . 5
33 fveq2 5687 . . . . 5
3432, 33oveq12d 6058 . . . 4
3526, 5, 30, 31, 34cbvmptf 24021 . . 3
36 simpr 448 . . . . . 6
375fvmpt2f 24025 . . . . . 6
3836, 2, 37syl2anc 643 . . . . 5
395fvmpt2f 24025 . . . . . 6
4036, 11, 39syl2anc 643 . . . . 5
4138, 40oveq12d 6058 . . . 4
421, 41mpteq2da 4254 . . 3
4335, 42syl5eq 2448 . 2
4425, 43eqtrd 2436 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wnf 1550   wceq 1649   wcel 1721  wnfc 2527  wral 2666   cmpt 4226   wfn 5408  cfv 5413  (class class class)co 6040   cof 6262 This theorem is referenced by:  esumaddf  24406 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264
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