Theorem hbco 2508

Description: Bound-variable hypothesis builder for function value.

Hypotheses

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

Assertion

Ref	Expression
hbco	⊢ (y ∈ (A ∘ B) → ∀x y ∈ (A ∘ B))

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

Proof of Theorem hbco

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . . 6 ⊢ (y ∈ z → ∀x y ∈ z)
2		hbco.2	. . . . . 6 ⊢ (y ∈ B → ∀x y ∈ B)
3		ax-17 925	. . . . . 6 ⊢ (y ∈ v → ∀x y ∈ v)
4	1, 2, 3	hbbr 2095	. . . . 5 ⊢ (zBv → ∀x zBv)
5		hbco.1	. . . . . 6 ⊢ (y ∈ A → ∀x y ∈ A)
6		ax-17 925	. . . . . 6 ⊢ (y ∈ w → ∀x y ∈ w)
7	3, 5, 6	hbbr 2095	. . . . 5 ⊢ (vAw → ∀x vAw)
8	4, 7	hban 704	. . . 4 ⊢ ((zBv ∧ vAw) → ∀x(zBv ∧ vAw))
9	8	hbex 701	. . 3 ⊢ (∃v(zBv ∧ vAw) → ∀x∃v(zBv ∧ vAw))
10	9	hbopab 2111	. 2 ⊢ (y ∈ {⟨z, w⟩∣∃v(zBv ∧ vAw)} → ∀x y ∈ {⟨z, w⟩∣∃v(zBv ∧ vAw)})
11		df-co 2427	. . 3 ⊢ (A ∘ B) = {⟨z, w⟩∣∃v(zBv ∧ vAw)}
12	11	eleq2i 1153	. 2 ⊢ (y ∈ (A ∘ B) ↔ y ∈ {⟨z, w⟩∣∃v(zBv ∧ vAw)})
13	12	bial 695	. 2 ⊢ (∀x y ∈ (A ∘ B) ↔ ∀x y ∈ {⟨z, w⟩∣∃v(zBv ∧ vAw)})
14	10, 12, 13	3imtr4 192	1 ⊢ (y ∈ (A ∘ B) → ∀x y ∈ (A ∘ B))

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   ∈ wel 803   ∈ wcel 1092   class class class wbr 2054  {copab 2055   ∘ ccom 2414

This theorem is referenced by:  hbfun 2684  ac6lem 3575

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-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-co 2427

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