Theorem hbima 2609

Description: Bound-variable hypothesis builder for image.

Hypotheses

Ref	Expression
hbima.1	⊢ (y ∈ A → ∀x y ∈ A)
hbima.2	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
hbima	⊢ (y ∈ (A “ B) → ∀x y ∈ (A “ B))

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

Proof of Theorem hbima

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . 5 ⊢ (w ∈ z → ∀x w ∈ z)
2		hbima.2	. . . . 5 ⊢ (y ∈ B → ∀x y ∈ B)
3	1, 2	hbel 1172	. . . 4 ⊢ (z ∈ B → ∀x z ∈ B)
4		ax-17 925	. . . . . 6 ⊢ (w ∈ y → ∀x w ∈ y)
5	1, 4	hbop 1879	. . . . 5 ⊢ (w ∈ ⟨z, y⟩ → ∀x w ∈ ⟨z, y⟩)
6		hbima.1	. . . . 5 ⊢ (y ∈ A → ∀x y ∈ A)
7	5, 6	hbel 1172	. . . 4 ⊢ (⟨z, y⟩ ∈ A → ∀x⟨z, y⟩ ∈ A)
8	3, 7	hban 704	. . 3 ⊢ ((z ∈ B ∧ ⟨z, y⟩ ∈ A) → ∀x(z ∈ B ∧ ⟨z, y⟩ ∈ A))
9	8	hbex 701	. 2 ⊢ (∃z(z ∈ B ∧ ⟨z, y⟩ ∈ A) → ∀x∃z(z ∈ B ∧ ⟨z, y⟩ ∈ A))
10		visset 1350	. . 3 ⊢ y ∈ V
11	10	elima3 2608	. 2 ⊢ (y ∈ (A “ B) ↔ ∃z(z ∈ B ∧ ⟨z, y⟩ ∈ A))
12	11	bial 695	. 2 ⊢ (∀x y ∈ (A “ B) ↔ ∀x∃z(z ∈ B ∧ ⟨z, y⟩ ∈ A))
13	9, 11, 12	3imtr4 192	1 ⊢ (y ∈ (A “ B) → ∀x y ∈ (A “ B))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803   ∈ wcel 1092  ⟨cop 1810   “ cima 2413

This theorem is referenced by:  hbfv 2837

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-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431

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