Theorem hbopr 3017

Description: Bound-variable hypothesis builder for operation value.

Hypotheses

Ref	Expression
hbopr.1	⊢ (y ∈ A → ∀x y ∈ A)
hbopr.2	⊢ (y ∈ F → ∀x y ∈ F)
hbopr.3	⊢ (y ∈ B → ∀x y ∈ B)

Assertion

Ref	Expression
hbopr	⊢ (y ∈ (AFB) → ∀x y ∈ (AFB))

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

Proof of Theorem hbopr

Step	Hyp	Ref	Expression
1		hbopr.2	. . 3 ⊢ (y ∈ F → ∀x y ∈ F)
2		hbopr.1	. . . 4 ⊢ (y ∈ A → ∀x y ∈ A)
3		hbopr.3	. . . 4 ⊢ (y ∈ B → ∀x y ∈ B)
4	2, 3	hbop 1879	. . 3 ⊢ (y ∈ ⟨A, B⟩ → ∀x y ∈ ⟨A, B⟩)
5	1, 4	hbfv 2837	. 2 ⊢ (y ∈ (F ‘⟨A, B⟩) → ∀x y ∈ (F ‘⟨A, B⟩))
6		df-opr 3003	. . 3 ⊢ (AFB) = (F ‘⟨A, B⟩)
7	6	eleq2i 1153	. 2 ⊢ (y ∈ (AFB) ↔ y ∈ (F ‘⟨A, B⟩))
8	7	bial 695	. 2 ⊢ (∀x y ∈ (AFB) ↔ ∀x y ∈ (F ‘⟨A, B⟩))
9	5, 7, 8	3imtr4 192	1 ⊢ (y ∈ (AFB) → ∀x y ∈ (AFB))

Syntax hints:   → wi 2  ∀wal 672   ∈ wcel 1092  ⟨cop 1810   ‘cfv 2422  (class class class)co 3001

This theorem is referenced by:  oawordeulem 3156  om2uzsuc 4652

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-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-opr 3003

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