Theorem hbcnv 2516

Description: Bound-variable hypothesis builder for converse.

Hypothesis

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

Assertion

Ref	Expression
hbcnv	|- (y e. `'A -> A.x y e. `'A)

Distinct variable group(s):   y,A   x,y

Proof of Theorem hbcnv

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 |- (y = <.z, w>. -> A.x y = <.z, w>.)
2		ax-17 925	. . . . . 6 |- (y e. w -> A.x y e. w)
3		hbcnv.1	. . . . . 6 |- (y e. A -> A.x y e. A)
4		ax-17 925	. . . . . 6 |- (y e. z -> A.x y e. z)
5	2, 3, 4	hbbr 2095	. . . . 5 |- (wAz -> A.x wAz)
6	1, 5	hban 704	. . . 4 |- ((y = <.z, w>. /\ wAz) -> A.x(y = <.z, w>. /\ wAz))
7	6	hbex 701	. . 3 |- (E.w(y = <.z, w>. /\ wAz) -> A.xE.w(y = <.z, w>. /\ wAz))
8	7	hbex 701	. 2 |- (E.zE.w(y = <.z, w>. /\ wAz) -> A.xE.zE.w(y = <.z, w>. /\ wAz))
9		elcnv 2514	. 2 |- (y e. `'A <-> E.zE.w(y = <.z, w>. /\ wAz))
10	9	bial 695	. 2 |- (A.x y e. `'A <-> A.xE.zE.w(y = <.z, w>. /\ wAz))
11	8, 9, 10	3imtr4 192	1 |- (y e. `'A -> A.x y e. `'A)

Syntax hints:   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   e. wel 803   = wceq 1091   e. wcel 1092  <.cop 1810   class class class wbr 2054  `'ccnv 2409

This theorem is referenced by:  hbdm 2565  hbfun 2684  hbf1 2779

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-cnv 2426

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