Theorem hbfn 3584

Description: Bound-variable hypothesis builder for a function with domain.

Hypotheses

Ref	Expression
hbfn.1	|- (y e. F -> A.x y e. F)
hbfn.2	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbfn	|- (F Fn A -> A.x F Fn A)

Distinct variable groups:   y,F   y,A   x,y

Proof of Theorem hbfn

Step	Hyp	Ref	Expression
1		hbfn.1	. . . 4 |- (y e. F -> A.x y e. F)
2	1	hbfun 3536	. . 3 |- (Fun F -> A.xFun F)
3	1	hbdm 3352	. . . 4 |- (y e. dom F -> A.x y e. dom F)
4		hbfn.2	. . . 4 |- (y e. A -> A.x y e. A)
5	3, 4	hbeq 1565	. . 3 |- (dom F = A -> A.xdom F = A)
6	2, 5	hban 1009	. 2 |- ((Fun F /\ dom F = A) -> A.x(Fun F /\ dom F = A))
7		df-fn 3193	. 2 |- (F Fn A <-> (Fun F /\ dom F = A))
8	7	albii 999	. 2 |- (A.x F Fn A <-> A.x(Fun F /\ dom F = A))
9	6, 7, 8	3imtr4 219	1 |- (F Fn A -> A.x F Fn A)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  dom cdm 3170  Fun wfun 3176   Fn wfn 3177

This theorem is referenced by:  fnopabg 3615  hbf 3625  hbfo 3671

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-dm 3188  df-rn 3189  df-fun 3192  df-fn 3193

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