Theorem hbfun 3536

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 3245	. . 3 |- (Rel F -> A.xRel F)
3	1	hbcnv 3295	. . . . 5 |- (y e. `'F -> A.x y e. `'F)
4	1, 3	hbco 3287	. . . 4 |- (y e. (F o. `'F) -> A.x y e. (F o. `'F))
5		ax-17 971	. . . 4 |- (y e. I -> A.x y e. I)
6	4, 5	hbss 2062	. . 3 |- ((F o. `'F) (_ I -> A.x(F o. `'F) (_ I)
7	2, 6	hban 1009	. 2 |- ((Rel F /\ (F o. `'F) (_ I) -> A.x(Rel F /\ (F o. `'F) (_ I))
8		df-fun 3192	. 2 |- (Fun F <-> (Rel F /\ (F o. `'F) (_ I))
9	8	albii 999	. 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 954   e. wcel 958   (_ wss 2047  Icid 2831  `'ccnv 3169   o. ccom 3174  Rel wrel 3175  Fun wfun 3176

This theorem is referenced by:  hbfn 3584  hbf1 3663

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-rel 3185  df-cnv 3186  df-co 3187  df-fun 3192

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