Theorem hbfvd 3736

Description: Deduction version of bound-variable hypothesis builder hbfv 3735. If a closed theorem version is desired, see hbfvd2 3737.

Hypotheses

Ref	Expression
hbfvd.1	⊢ (φ → ∀xφ)
hbfvd.2	⊢ (φ → (y ∈ F → ∀x y ∈ F))
hbfvd.3	⊢ (φ → (y ∈ A → ∀x y ∈ A))

Assertion

Ref	Expression
hbfvd	⊢ (φ → (y ∈ (F ‘A) → ∀x y ∈ (F ‘A)))

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

Proof of Theorem hbfvd

Step	Hyp	Ref	Expression
1		hba1 1005	. . . . 5 ⊢ (∀x z ∈ F → ∀x∀x z ∈ F)
2	1	hbab 1470	. . . 4 ⊢ (y ∈ {z∣∀x z ∈ F} → ∀x y ∈ {z∣∀x z ∈ F})
3		hba1 1005	. . . . 5 ⊢ (∀x z ∈ A → ∀x∀x z ∈ A)
4	3	hbab 1470	. . . 4 ⊢ (y ∈ {z∣∀x z ∈ A} → ∀x y ∈ {z∣∀x z ∈ A})
5	2, 4	hbfv 3735	. . 3 ⊢ (y ∈ ({z∣∀x z ∈ F} ‘{z∣∀x z ∈ A}) → ∀x y ∈ ({z∣∀x z ∈ F} ‘{z∣∀x z ∈ A}))
6	5	a1i 8	. 2 ⊢ (φ → (y ∈ ({z∣∀x z ∈ F} ‘{z∣∀x z ∈ A}) → ∀x y ∈ ({z∣∀x z ∈ F} ‘{z∣∀x z ∈ A})))
7		hbfvd.2	. . . . . . 7 ⊢ (φ → (y ∈ F → ∀x y ∈ F))
8	7	19.21aiv 1288	. . . . . 6 ⊢ (φ → ∀y(y ∈ F → ∀x y ∈ F))
9		abidhb 1915	. . . . . 6 ⊢ (∀y(y ∈ F → ∀x y ∈ F) → {z∣∀x z ∈ F} = F)
10	8, 9	syl 10	. . . . 5 ⊢ (φ → {z∣∀x z ∈ F} = F)
11	10	fveq1d 3732	. . . 4 ⊢ (φ → ({z∣∀x z ∈ F} ‘{z∣∀x z ∈ A}) = (F ‘{z∣∀x z ∈ A}))
12		hbfvd.3	. . . . . . 7 ⊢ (φ → (y ∈ A → ∀x y ∈ A))
13	12	19.21aiv 1288	. . . . . 6 ⊢ (φ → ∀y(y ∈ A → ∀x y ∈ A))
14		abidhb 1915	. . . . . 6 ⊢ (∀y(y ∈ A → ∀x y ∈ A) → {z∣∀x z ∈ A} = A)
15	13, 14	syl 10	. . . . 5 ⊢ (φ → {z∣∀x z ∈ A} = A)
16	15	fveq2d 3734	. . . 4 ⊢ (φ → (F ‘{z∣∀x z ∈ A}) = (F ‘A))
17	11, 16	eqtrd 1510	. . 3 ⊢ (φ → ({z∣∀x z ∈ F} ‘{z∣∀x z ∈ A}) = (F ‘A))
18	17	eleq2d 1544	. 2 ⊢ (φ → (y ∈ ({z∣∀x z ∈ F} ‘{z∣∀x z ∈ A}) ↔ y ∈ (F ‘A)))
19		hbfvd.1	. . 3 ⊢ (φ → ∀xφ)
20	19, 18	albid 1106	. 2 ⊢ (φ → (∀x y ∈ ({z∣∀x z ∈ F} ‘{z∣∀x z ∈ A}) ↔ ∀x y ∈ (F ‘A)))
21	6, 18, 20	3imtr3d 544	1 ⊢ (φ → (y ∈ (F ‘A) → ∀x y ∈ (F ‘A)))

Syntax hints:   → wi 3  ∀wal 956   = wceq 958   ∈ wcel 960  {cab 1466   ‘cfv 3188

This theorem is referenced by:  csbfv12g 3748

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-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-cnv 3192  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fv 3204

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