Theorem hbfvd 3725

Description: Deduction version of bound-variable hypothesis builder hbfv 3724. If a closed theorem version is desired, see hbfvd2 3726.

Hypotheses

Ref	Expression
hbfvd.1	|- (ph -> A.xph)
hbfvd.2	|- (ph -> (y e. F -> A.x y e. F))
hbfvd.3	|- (ph -> (y e. A -> A.x y e. A))

Assertion

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

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

Proof of Theorem hbfvd

Step	Hyp	Ref	Expression
1		hba1 1002	. . . . 5 |- (A.x z e. F -> A.xA.x z e. F)
2	1	hbab 1466	. . . 4 |- (y e. {z | A.x z e. F} -> A.x y e. {z | A.x z e. F})
3		hba1 1002	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
4	3	hbab 1466	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
5	2, 4	hbfv 3724	. . 3 |- (y e. ({z | A.x z e. F}` {z | A.x z e. A}) -> A.x y e. ({z | A.x z e. F}` {z | A.x z e. A}))
6	5	a1i 8	. 2 |- (ph -> (y e. ({z | A.x z e. F}` {z | A.x z e. A}) -> A.x y e. ({z | A.x z e. F}` {z | A.x z e. A})))
7		hbfvd.2	. . . . . . 7 |- (ph -> (y e. F -> A.x y e. F))
8	7	19.21aiv 1285	. . . . . 6 |- (ph -> A.y(y e. F -> A.x y e. F))
9		abidhb 1909	. . . . . 6 |- (A.y(y e. F -> A.x y e. F) -> {z | A.x z e. F} = F)
10	8, 9	syl 10	. . . . 5 |- (ph -> {z | A.x z e. F} = F)
11	10	fveq1d 3721	. . . 4 |- (ph -> ({z | A.x z e. F}` {z | A.x z e. A}) = (F` {z | A.x z e. A}))
12		hbfvd.3	. . . . . . 7 |- (ph -> (y e. A -> A.x y e. A))
13	12	19.21aiv 1285	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
14		abidhb 1909	. . . . . 6 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
15	13, 14	syl 10	. . . . 5 |- (ph -> {z | A.x z e. A} = A)
16	15	fveq2d 3723	. . . 4 |- (ph -> (F` {z | A.x z e. A}) = (F` A))
17	11, 16	eqtrd 1505	. . 3 |- (ph -> ({z | A.x z e. F}` {z | A.x z e. A}) = (F` A))
18	17	eleq2d 1539	. 2 |- (ph -> (y e. ({z | A.x z e. F}` {z | A.x z e. A}) <-> y e. (F` A)))
19		hbfvd.1	. . 3 |- (ph -> A.xph)
20	19, 18	albid 1103	. 2 |- (ph -> (A.x y e. ({z | A.x z e. F}` {z | A.x z e. A}) <-> A.x y e. (F` A)))
21	6, 18, 20	3imtr3d 541	1 |- (ph -> (y e. (F` A) -> A.x y e. (F` A)))

Syntax hints:   -> wi 3  A.wal 953   = wceq 955   e. wcel 957  {cab 1462  ` cfv 3178

This theorem is referenced by:  csbfv12g 3737

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-xp 3180  df-cnv 3182  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fv 3194

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