Metamath Proof Explorer


Theorem hsmexlem6

Description: Lemma for hsmex . (Contributed by Stefan O'Rear, 14-Feb-2015)

Ref Expression
Hypotheses hsmexlem4.x 𝑋 ∈ V
hsmexlem4.h 𝐻 = ( rec ( ( 𝑧 ∈ V ↦ ( har ‘ 𝒫 ( 𝑋 × 𝑧 ) ) ) , ( har ‘ 𝒫 𝑋 ) ) ↾ ω )
hsmexlem4.u 𝑈 = ( 𝑥 ∈ V ↦ ( rec ( ( 𝑦 ∈ V ↦ 𝑦 ) , 𝑥 ) ↾ ω ) )
hsmexlem4.s 𝑆 = { 𝑎 ( 𝑅1 “ On ) ∣ ∀ 𝑏 ∈ ( TC ‘ { 𝑎 } ) 𝑏𝑋 }
hsmexlem4.o 𝑂 = OrdIso ( E , ( rank “ ( ( 𝑈𝑑 ) ‘ 𝑐 ) ) )
Assertion hsmexlem6 𝑆 ∈ V

Proof

Step Hyp Ref Expression
1 hsmexlem4.x 𝑋 ∈ V
2 hsmexlem4.h 𝐻 = ( rec ( ( 𝑧 ∈ V ↦ ( har ‘ 𝒫 ( 𝑋 × 𝑧 ) ) ) , ( har ‘ 𝒫 𝑋 ) ) ↾ ω )
3 hsmexlem4.u 𝑈 = ( 𝑥 ∈ V ↦ ( rec ( ( 𝑦 ∈ V ↦ 𝑦 ) , 𝑥 ) ↾ ω ) )
4 hsmexlem4.s 𝑆 = { 𝑎 ( 𝑅1 “ On ) ∣ ∀ 𝑏 ∈ ( TC ‘ { 𝑎 } ) 𝑏𝑋 }
5 hsmexlem4.o 𝑂 = OrdIso ( E , ( rank “ ( ( 𝑈𝑑 ) ‘ 𝑐 ) ) )
6 fvex ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ) ∈ V
7 1 2 3 4 5 hsmexlem5 ( 𝑑𝑆 → ( rank ‘ 𝑑 ) ∈ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) )
8 4 ssrab3 𝑆 ( 𝑅1 “ On )
9 8 sseli ( 𝑑𝑆𝑑 ( 𝑅1 “ On ) )
10 harcl ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ∈ On
11 r1fnon 𝑅1 Fn On
12 11 fndmi dom 𝑅1 = On
13 10 12 eleqtrri ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ∈ dom 𝑅1
14 rankr1ag ( ( 𝑑 ( 𝑅1 “ On ) ∧ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ∈ dom 𝑅1 ) → ( 𝑑 ∈ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ) ↔ ( rank ‘ 𝑑 ) ∈ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ) )
15 9 13 14 sylancl ( 𝑑𝑆 → ( 𝑑 ∈ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ) ↔ ( rank ‘ 𝑑 ) ∈ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ) )
16 7 15 mpbird ( 𝑑𝑆𝑑 ∈ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) ) )
17 16 ssriv 𝑆 ⊆ ( 𝑅1 ‘ ( har ‘ 𝒫 ( ω × ran 𝐻 ) ) )
18 6 17 ssexi 𝑆 ∈ V