Theorem hbimad 3418

Description: Deduction version of bound-variable hypothesis builder hbima 3417. (Contributed by FL, 15-Dec-2006.)

Hypotheses

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

Assertion

Ref	Expression
hbimad	|- (ph -> (y e. (A"B) -> A.x y e. (A"B)))

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

Proof of Theorem hbimad

Step	Hyp	Ref	Expression
1		hba1 1005	. . . . 5 |- (A.x z e. A -> A.xA.x z e. A)
2	1	hbab 1470	. . . 4 |- (y e. {z | A.x z e. A} -> A.x y e. {z | A.x z e. A})
3		hba1 1005	. . . . 5 |- (A.x z e. B -> A.xA.x z e. B)
4	3	hbab 1470	. . . 4 |- (y e. {z | A.x z e. B} -> A.x y e. {z | A.x z e. B})
5	2, 4	hbima 3417	. . 3 |- (y e. ({z | A.x z e. A}"{z | A.x z e. B}) -> A.x y e. ({z | A.x z e. A}"{z | A.x z e. B}))
6	5	a1i 8	. 2 |- (ph -> (y e. ({z | A.x z e. A}"{z | A.x z e. B}) -> A.x y e. ({z | A.x z e. A}"{z | A.x z e. B})))
7		hbimad.2	. . . . . . 7 |- (ph -> (y e. A -> A.x y e. A))
8	7	19.21aiv 1288	. . . . . 6 |- (ph -> A.y(y e. A -> A.x y e. A))
9		abidhb 1915	. . . . . 6 |- (A.y(y e. A -> A.x y e. A) -> {z | A.x z e. A} = A)
10	8, 9	syl 10	. . . . 5 |- (ph -> {z | A.x z e. A} = A)
11	10	imaeq1d 3409	. . . 4 |- (ph -> ({z | A.x z e. A}"{z | A.x z e. B}) = (A"{z | A.x z e. B}))
12		hbimad.3	. . . . . . 7 |- (ph -> (y e. B -> A.x y e. B))
13	12	19.21aiv 1288	. . . . . 6 |- (ph -> A.y(y e. B -> A.x y e. B))
14		abidhb 1915	. . . . . 6 |- (A.y(y e. B -> A.x y e. B) -> {z | A.x z e. B} = B)
15	13, 14	syl 10	. . . . 5 |- (ph -> {z | A.x z e. B} = B)
16	15	imaeq2d 3410	. . . 4 |- (ph -> (A"{z | A.x z e. B}) = (A"B))
17	11, 16	eqtrd 1510	. . 3 |- (ph -> ({z | A.x z e. A}"{z | A.x z e. B}) = (A"B))
18	17	eleq2d 1544	. 2 |- (ph -> (y e. ({z | A.x z e. A}"{z | A.x z e. B}) <-> y e. (A"B)))
19		hbimad.1	. . 3 |- (ph -> A.xph)
20	19, 18	albid 1106	. 2 |- (ph -> (A.x y e. ({z | A.x z e. A}"{z | A.x z e. B}) <-> A.x y e. (A"B)))
21	6, 18, 20	3imtr3d 544	1 |- (ph -> (y e. (A"B) -> A.x y e. (A"B)))

Syntax hints:   -> wi 3  A.wal 956   = wceq 958   e. wcel 960  {cab 1466  "cima 3179

This theorem is referenced by:  csbima12g 3419

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197

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