Theorem hbfn 3580

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 3532	. . 3 |- (Fun F -> A.xFun F)
3	1	hbdm 3348	. . . 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 1563	. . 3 |- (dom F = A -> A.xdom F = A)
6	2, 5	hban 1008	. 2 |- ((Fun F /\ dom F = A) -> A.x(Fun F /\ dom F = A))
7		df-fn 3189	. 2 |- (F Fn A <-> (Fun F /\ dom F = A))
8	7	albii 998	. 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 953   = wceq 955   e. wcel 957  dom cdm 3166  Fun wfun 3172   Fn wfn 3173

This theorem is referenced by:  fnopabg 3611  hbf 3621  hbfo 3666

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-fun 3188  df-fn 3189

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