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Metamath Proof Explorer

Theorem wuntpos

Description: A weak universe is closed under transposition. (Contributed by Mario Carneiro, 12-Jan-2017)

Ref Expression
Hypotheses wun0.1 ⊢ φ → U ∈ WUni
wunop.2 ⊢ φ → A ∈ U
Assertion wuntpos ⊢ φ → tpos A ∈ U

Proof

Step Hyp Ref Expression
1 wun0.1 ⊢ φ → U ∈ WUni
2 wunop.2 ⊢ φ → A ∈ U
3 1 2 wundm ⊢ φ → dom ⁡ A ∈ U
4 1 3 wuncnv ⊢ φ → dom ⁡ A -1 ∈ U
5 1 wun0 ⊢ φ → ∅ ∈ U
6 1 5 wunsn ⊢ φ → ∅ ∈ U
7 1 4 6 wunun ⊢ φ → dom ⁡ A -1 ∪ ∅ ∈ U
8 1 2 wunrn ⊢ φ → ran ⁡ A ∈ U
9 1 7 8 wunxp ⊢ φ → dom ⁡ A -1 ∪ ∅ × ran ⁡ A ∈ U
10 tposssxp ⊢ tpos A ⊆ dom ⁡ A -1 ∪ ∅ × ran ⁡ A
11 10 a1i ⊢ φ → tpos A ⊆ dom ⁡ A -1 ∪ ∅ × ran ⁡ A
12 1 9 11 wunss ⊢ φ → tpos A ∈ U