Theorem hbf 3625

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 3584	. . 3 |- (F Fn A -> A.x F Fn A)
4	1	hbrn 3351	. . . 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 2062	. . 3 |- (ran F (_ B -> A.xran F (_ B)
7	3, 6	hban 1009	. 2 |- ((F Fn A /\ ran F (_ B) -> A.x(F Fn A /\ ran F (_ B))
8		df-f 3194	. 2 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
9	8	albii 999	. 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 954   e. wcel 958   (_ wss 2047  ran crn 3171   Fn wfn 3177  -->wf 3178

This theorem is referenced by:  hbf1 3663  fopab2 3823

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  df-f 3194

Copyright terms: Public domain