Theorem hbf 3631

Description: Bound-variable hypothesis builder for a mapping.

Hypotheses

Ref	Expression
hbf.1	|- (y e. F -> A.x y e. F)
hbf.2	|- (y e. A -> A.x y e. A)
hbf.3	|- (y e. B -> A.x y e. B)

Assertion

Ref	Expression
hbf	|- (F:A-->B -> A.x F:A-->B)

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

Proof of Theorem hbf

Step	Hyp	Ref	Expression
1		hbf.1	. . . 4 |- (y e. F -> A.x y e. F)
2		hbf.2	. . . 4 |- (y e. A -> A.x y e. A)
3	1, 2	hbfn 3590	. . 3 |- (F Fn A -> A.x F Fn A)
4	1	hbrn 3357	. . . 4 |- (y e. ran F -> A.x y e. ran F)
5		hbf.3	. . . 4 |- (y e. B -> A.x y e. B)
6	4, 5	hbss 2065	. . 3 |- (ran F (_ B -> A.xran F (_ B)
7	3, 6	hban 1011	. 2 |- ((F Fn A /\ ran F (_ B) -> A.x(F Fn A /\ ran F (_ B))
8		df-f 3200	. 2 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
9	8	albii 1001	. 2 |- (A.x F:A-->B <-> A.x(F Fn A /\ ran F (_ B))
10	7, 8, 9	3imtr4 219	1 |- (F:A-->B -> A.x F:A-->B)

Syntax hints:   -> wi 3   /\ wa 223  A.wal 956   e. wcel 960   (_ wss 2050  ran crn 3177   Fn wfn 3183  -->wf 3184

This theorem is referenced by:  hbf1 3669  fopab2 3829

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-dm 3194  df-rn 3195  df-fun 3198  df-fn 3199  df-f 3200

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