Theorem hbfv 2837

Description: Bound-variable hypothesis builder for function value.

Hypotheses

Ref	Expression
hbfv.1	|- (y e. F -> A.x y e. F)
hbfv.2	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
hbfv	|- (y e. (F` A) -> A.x y e. (F` A))

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

Proof of Theorem hbfv

Step	Hyp	Ref	Expression
1		hbfv.1	. . . . . 6 |- (y e. F -> A.x y e. F)
2		hbfv.2	. . . . . . 7 |- (y e. A -> A.x y e. A)
3	2	hbsn 1833	. . . . . 6 |- (y e. {A} -> A.x y e. {A})
4	1, 3	hbima 2609	. . . . 5 |- (y e. (F"{A}) -> A.x y e. (F"{A}))
5		ax-17 925	. . . . 5 |- (y e. {z} -> A.x y e. {z})
6	4, 5	hbeq 1171	. . . 4 |- ((F"{A}) = {z} -> A.x(F"{A}) = {z})
7	6	hbab 1096	. . 3 |- (y e. {z | (F"{A}) = {z}} -> A.x y e. {z | (F"{A}) = {z}})
8	7	hbuni 1925	. 2 |- (y e. U.{z | (F"{A}) = {z}} -> A.x y e. U.{z | (F"{A}) = {z}})
9		df-fv 2438	. . 3 |- (F` A) = U.{z | (F"{A}) = {z}}
10	9	eleq2i 1153	. 2 |- (y e. (F` A) <-> y e. U.{z | (F"{A}) = {z}})
11	10	bial 695	. 2 |- (A.x y e. (F` A) <-> A.x y e. U.{z | (F"{A}) = {z}})
12	8, 10, 11	3imtr4 192	1 |- (y e. (F` A) -> A.x y e. (F` A))

Syntax hints:   -> wi 2  A.wal 672  {cab 1090   = wceq 1091   e. wcel 1092  {csn 1808  U.cuni 1919  "cima 2413  ` cfv 2422

This theorem is referenced by:  fvopab2 2878  cleqfvf 2881  elrnopab 2884  abrexexlem2 2911  f1fvf 2917  hbiso 2930  hbrdg 2974  rdgsucopab 2984  rdgsucopabn 2985  frsucopab 2992  abianfplem 2999  hbopr 3017  elrnoprab 3054  dom2d 3307  unblem2 3432  unblem3 3433  inf0 3457  trcl 3489  tz9.12lem3 3505  rankid 3516  ranklon 3540  cardprc 3667  cardaleph 3690  om2uzsuc 4652

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-rex 1206  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-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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