Theorem fneq1 2718

Description: Equality theorem for function predicate with domain.

Assertion

Ref	Expression
fneq1	|- (F = G -> (F Fn A <-> G Fn A))

Proof of Theorem fneq1

Step	Hyp	Ref	Expression
1		funeq 2683	. . 3 |- (F = G -> (Fun F <-> Fun G))
2		dmeq 2531	. . . 4 |- (F = G -> dom F = dom G)
3	2	cleq1d 1109	. . 3 |- (F = G -> (dom F = A <-> dom G = A))
4	1, 3	anbi12d 476	. 2 |- (F = G -> ((Fun F /\ dom F = A) <-> (Fun G /\ dom G = A)))
5		df-fn 2433	. 2 |- (F Fn A <-> (Fun F /\ dom F = A))
6		df-fn 2433	. 2 |- (G Fn A <-> (Fun G /\ dom G = A))
7	4, 5, 6	3bitr4g 428	1 |- (F = G -> (F Fn A <-> G Fn A))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   = wceq 1091  dom cdm 2410  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  fn0 2739  fnopabg 2745  feq1 2748  foeq1 2784  f1oun 2815  f1o00 2823  f1osn 2827  fnopabfv 2858  tfrlem3 2951  tfrlem12 2960  abianfp 3000  fnoprab2 3039  en2d 3303  pw2en 3348  mapxpen 3390  unblem4 3434  inf3lem6 3469  r1fnon 3494  aceq3lem 3555  aceq4 3557  alephfnon 3668  om2uzran 4655  om2uzf1o 4656

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-fun 2432  df-fn 2433

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