This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem funcestrcsetclem3

Description: Lemma 3 for funcestrcsetc . (Contributed by AV, 22-Mar-2020)

Ref Expression
Hypotheses funcestrcsetc.e ⊢ E = ExtStrCat ⁡ U
funcestrcsetc.s ⊢ S = SetCat ⁡ U
funcestrcsetc.b ⊢ B = Base E
funcestrcsetc.c ⊢ C = Base S
funcestrcsetc.u ⊢ φ → U ∈ WUni
funcestrcsetc.f ⊢ φ → F = x ∈ B ⟼ Base x
Assertion funcestrcsetclem3 ⊢ φ → F : B ⟶ C

Proof

Step Hyp Ref Expression
1 funcestrcsetc.e ⊢ E = ExtStrCat ⁡ U
2 funcestrcsetc.s ⊢ S = SetCat ⁡ U
3 funcestrcsetc.b ⊢ B = Base E
4 funcestrcsetc.c ⊢ C = Base S
5 funcestrcsetc.u ⊢ φ → U ∈ WUni
6 funcestrcsetc.f ⊢ φ → F = x ∈ B ⟼ Base x
7 1 3 5 estrcbasbas ⊢ φ ∧ x ∈ B → Base x ∈ U
8 2 5 setcbas ⊢ φ → U = Base S
9 8 eqcomd ⊢ φ → Base S = U
10 9 adantr ⊢ φ ∧ x ∈ B → Base S = U
11 7 10 eleqtrrd ⊢ φ ∧ x ∈ B → Base x ∈ Base S
12 11 4 eleqtrrdi ⊢ φ ∧ x ∈ B → Base x ∈ C
13 6 12 fmpt3d ⊢ φ → F : B ⟶ C