Theorem hbfun 3542

Description: Bound-variable hypothesis builder for a function.

Hypothesis

Ref	Expression
hbfun.1	|- (y e. F -> A.x y e. F)

Assertion

Ref	Expression
hbfun	|- (Fun F -> A.xFun F)

Distinct variable groups:   y,F   x,y

Proof of Theorem hbfun

Step	Hyp	Ref	Expression
1		hbfun.1	. . . 4 |- (y e. F -> A.x y e. F)
2	1	hbrel 3251	. . 3 |- (Rel F -> A.xRel F)
3	1	hbcnv 3301	. . . . 5 |- (y e. `'F -> A.x y e. `'F)
4	1, 3	hbco 3293	. . . 4 |- (y e. (F o. `'F) -> A.x y e. (F o. `'F))
5		ax-17 973	. . . 4 |- (y e. I -> A.x y e. I)
6	4, 5	hbss 2065	. . 3 |- ((F o. `'F) (_ I -> A.x(F o. `'F) (_ I)
7	2, 6	hban 1011	. 2 |- ((Rel F /\ (F o. `'F) (_ I) -> A.x(Rel F /\ (F o. `'F) (_ I))
8		df-fun 3198	. 2 |- (Fun F <-> (Rel F /\ (F o. `'F) (_ I))
9	8	albii 1001	. 2 |- (A.xFun F <-> A.x(Rel F /\ (F o. `'F) (_ I))
10	7, 8, 9	3imtr4 219	1 |- (Fun F -> A.xFun F)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   e. wcel 960   (_ wss 2050  Icid 2837  `'ccnv 3175   o. ccom 3180  Rel wrel 3181  Fun wfun 3182

This theorem is referenced by:  hbfn 3590  hbf1 3669

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-rel 3191  df-cnv 3192  df-co 3193  df-fun 3198

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