Theorem hbrn 3345

Description: Bound-variable hypothesis builder for range.

Hypothesis

Ref	Expression
hbrn.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbrn	|- (y e. ran A -> A.x y e. ran A)

Distinct variable groups:   x,y   y,A

Proof of Theorem hbrn

Step	Hyp	Ref	Expression
1		ax-17 969	. . . 4 |- (w e. <.z, y>. -> A.x w e. <.z, y>.)
2		ax-17 969	. . . . 5 |- (y e. z -> A.x y e. z)
3		hbrn.1	. . . . 5 |- (y e. A -> A.x y e. A)
4	2, 3	hbel 1563	. . . 4 |- (z e. A -> A.x z e. A)
5	1, 4	hbel 1563	. . 3 |- (<.z, y>. e. A -> A.x<.z, y>. e. A)
6	5	hbex 1004	. 2 |- (E.z<.z, y>. e. A -> A.xE.z<.z, y>. e. A)
7		visset 1809	. . 3 |- y e. V
8	7	elrn2 3343	. 2 |- (y e. ran A <-> E.z<.z, y>. e. A)
9	8	albii 997	. 2 |- (A.x y e. ran A <-> A.xE.z<.z, y>. e. A)
10	6, 8, 9	3imtr4 219	1 |- (y e. ran A -> A.x y e. ran A)

Syntax hints:   -> wi 3  A.wal 952   e. wcel 956  E.wex 978  <.cop 2407  ran crn 3166

This theorem is referenced by:  hbdm 3346  zfrep6 3606  hbf 3617  hbfo 3662  fopab2 3814  rnssopab 3816  fopabco 3823  ac6lem 4734

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-cnv 3181  df-dm 3183  df-rn 3184

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