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Theorem off2 27937
 Description: The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
Hypotheses
Ref Expression
off2.1
off2.2
off2.3
off2.4
off2.5
off2.6
Assertion
Ref Expression
off2
Distinct variable groups:   ,   ,,   ,,   ,,   ,,   ,,   ,,
Allowed substitution hints:   (,)   (,)   (,)   ()   (,)   (,)

Proof of Theorem off2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 off2.2 . . . . . 6
21adantr 465 . . . . 5
3 off2.6 . . . . . . 7
4 inss1 3661 . . . . . . 7
53, 4syl6eqssr 3495 . . . . . 6
65sselda 3444 . . . . 5
72, 6ffvelrnd 6012 . . . 4
8 off2.3 . . . . . 6
98adantr 465 . . . . 5
10 inss2 3662 . . . . . . 7
113, 10syl6eqssr 3495 . . . . . 6
1211sselda 3444 . . . . 5
139, 12ffvelrnd 6012 . . . 4
14 off2.1 . . . . . 6
1514ralrimivva 2827 . . . . 5
1615adantr 465 . . . 4
17 ovrspc2v 6302 . . . 4
187, 13, 16, 17syl21anc 1231 . . 3
19 eqid 2404 . . 3
2018, 19fmptd 6035 . 2
21 ffn 5716 . . . . . 6
221, 21syl 17 . . . . 5
23 ffn 5716 . . . . . 6
248, 23syl 17 . . . . 5
25 off2.4 . . . . 5
26 off2.5 . . . . 5
27 eqid 2404 . . . . 5
28 eqidd 2405 . . . . 5
29 eqidd 2405 . . . . 5
3022, 24, 25, 26, 27, 28, 29offval 6530 . . . 4
313mpteq1d 4478 . . . 4
3230, 31eqtrd 2445 . . 3
3332feq1d 5702 . 2
3420, 33mpbird 234 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1407   wcel 1844  wral 2756   cin 3415   cmpt 4455   wfn 5566  wf 5567  cfv 5571  (class class class)co 6280   cof 6521 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pr 4632 This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-of 6523 This theorem is referenced by: (None)
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