Theorem funssfv 2841

Description: The value of a member of the domain of a subclass of a function.

Assertion

Ref	Expression
funssfv	|- (((Fun F /\ G (_ F) /\ A e. dom G) -> (F` A) = (G` A))

Proof of Theorem funssfv

Step	Hyp	Ref	Expression
1		funssres 2698	. . . . 5 |- ((Fun F /\ G (_ F) -> (F |` dom G) = G)
2	1	fveq1d 2834	. . . 4 |- ((Fun F /\ G (_ F) -> ((F |` dom G)` A) = (G` A))
3	2	cleqcomd 1106	. . 3 |- ((Fun F /\ G (_ F) -> (G` A) = ((F |` dom G)` A))
4		fvres 2840	. . 3 |- (A e. dom G -> ((F |` dom G)` A) = (F` A))
5	3, 4	sylan9eq 1144	. 2 |- (((Fun F /\ G (_ F) /\ A e. dom G) -> (G` A) = (F` A))
6	5	cleqcomd 1106	1 |- (((Fun F /\ G (_ F) /\ A e. dom G) -> (F` A) = (G` A))

Syntax hints:   -> wi 2   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487  dom cdm 2410   |` cres 2412  Fun wfun 2416  ` cfv 2422

This theorem is referenced by:  tfrlem9 2957  tfrlem11 2959

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-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438

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