Metamath Proof Explorer

Theorem bnj1234

Description: Technical lemma for bnj60 . This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

Ref Expression
Hypotheses bnj1234.2 ⊢ Y = x f ↾ pred x A R
bnj1234.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
bnj1234.4 ⊢ Z = x g ↾ pred x A R
bnj1234.5 ⊢ D = g | ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ Z
Assertion bnj1234 ⊢ C = D

Proof

Step Hyp Ref Expression
1 bnj1234.2 ⊢ Y = x f ↾ pred x A R
2 bnj1234.3 ⊢ C = f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y
3 bnj1234.4 ⊢ Z = x g ↾ pred x A R
4 bnj1234.5 ⊢ D = g | ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ Z
5 fneq1 ⊢ f = g → f Fn d ↔ g Fn d
6 fveq1 ⊢ f = g → f ⁡ x = g ⁡ x
7 reseq1 ⊢ f = g → f ↾ pred x A R = g ↾ pred x A R
8 7 opeq2d ⊢ f = g → x f ↾ pred x A R = x g ↾ pred x A R
9 8 1 3 3eqtr4g ⊢ f = g → Y = Z
10 9 fveq2d ⊢ f = g → G ⁡ Y = G ⁡ Z
11 6 10 eqeq12d ⊢ f = g → f ⁡ x = G ⁡ Y ↔ g ⁡ x = G ⁡ Z
12 11 ralbidv ⊢ f = g → ∀ x ∈ d f ⁡ x = G ⁡ Y ↔ ∀ x ∈ d g ⁡ x = G ⁡ Z
13 5 12 anbi12d ⊢ f = g → f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y ↔ g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ Z
14 13 rexbidv ⊢ f = g → ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y ↔ ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ Z
15 14 cbvabv ⊢ f | ∃ d ∈ B f Fn d ∧ ∀ x ∈ d f ⁡ x = G ⁡ Y = g | ∃ d ∈ B g Fn d ∧ ∀ x ∈ d g ⁡ x = G ⁡ Z
16 15 2 4 3eqtr4i ⊢ C = D