Metamath Proof Explorer


Theorem mnurndlem2

Description: Lemma for mnurnd . Deduction theorem input. (Contributed by Rohan Ridenour, 13-Aug-2023)

Ref Expression
Hypotheses mnurndlem2.1 𝑀 = { 𝑘 ∣ ∀ 𝑙𝑘 ( 𝒫 𝑙𝑘 ∧ ∀ 𝑚𝑛𝑘 ( 𝒫 𝑙𝑛 ∧ ∀ 𝑝𝑙 ( ∃ 𝑞𝑘 ( 𝑝𝑞𝑞𝑚 ) → ∃ 𝑟𝑚 ( 𝑝𝑟 𝑟𝑛 ) ) ) ) }
mnurndlem2.2 ( 𝜑𝑈𝑀 )
mnurndlem2.3 ( 𝜑𝐴𝑈 )
mnurndlem2.4 ( 𝜑𝐹 : 𝐴𝑈 )
mnurndlem2.5 𝐴 ∈ V
Assertion mnurndlem2 ( 𝜑 → ran 𝐹𝑈 )

Proof

Step Hyp Ref Expression
1 mnurndlem2.1 𝑀 = { 𝑘 ∣ ∀ 𝑙𝑘 ( 𝒫 𝑙𝑘 ∧ ∀ 𝑚𝑛𝑘 ( 𝒫 𝑙𝑛 ∧ ∀ 𝑝𝑙 ( ∃ 𝑞𝑘 ( 𝑝𝑞𝑞𝑚 ) → ∃ 𝑟𝑚 ( 𝑝𝑟 𝑟𝑛 ) ) ) ) }
2 mnurndlem2.2 ( 𝜑𝑈𝑀 )
3 mnurndlem2.3 ( 𝜑𝐴𝑈 )
4 mnurndlem2.4 ( 𝜑𝐹 : 𝐴𝑈 )
5 mnurndlem2.5 𝐴 ∈ V
6 2 adantr ( ( 𝜑𝑎𝐴 ) → 𝑈𝑀 )
7 3 adantr ( ( 𝜑𝑎𝐴 ) → 𝐴𝑈 )
8 simpr ( ( 𝜑𝑎𝐴 ) → 𝑎𝐴 )
9 1 6 7 8 mnutrcld ( ( 𝜑𝑎𝐴 ) → 𝑎𝑈 )
10 4 ffvelrnda ( ( 𝜑𝑎𝐴 ) → ( 𝐹𝑎 ) ∈ 𝑈 )
11 1 6 10 7 mnuprd ( ( 𝜑𝑎𝐴 ) → { ( 𝐹𝑎 ) , 𝐴 } ∈ 𝑈 )
12 1 6 9 11 mnuprd ( ( 𝜑𝑎𝐴 ) → { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∈ 𝑈 )
13 12 ralrimiva ( 𝜑 → ∀ 𝑎𝐴 { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∈ 𝑈 )
14 eqid ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) = ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } )
15 14 rnmptss ( ∀ 𝑎𝐴 { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ∈ 𝑈 → ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ⊆ 𝑈 )
16 13 15 syl ( 𝜑 → ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ⊆ 𝑈 )
17 1 2 3 16 mnuop3d ( 𝜑 → ∃ 𝑤𝑈𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) )
18 simprl ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → 𝑤𝑈 )
19 sseq2 ( 𝑏 = 𝑤 → ( ran 𝐹𝑏 ↔ ran 𝐹𝑤 ) )
20 19 adantl ( ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) ) ) ∧ 𝑏 = 𝑤 ) → ( ran 𝐹𝑏 ↔ ran 𝐹𝑤 ) )
21 4 adantr ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → 𝐹 : 𝐴𝑈 )
22 simprr ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) )
23 21 5 22 mnurndlem1 ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ran 𝐹𝑤 )
24 18 20 23 rspcedvd ( ( 𝜑 ∧ ( 𝑤𝑈 ∧ ∀ 𝑖𝐴 ( ∃ 𝑣 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) 𝑖𝑣 → ∃ 𝑢 ∈ ran ( 𝑎𝐴 ↦ { 𝑎 , { ( 𝐹𝑎 ) , 𝐴 } } ) ( 𝑖𝑢 𝑢𝑤 ) ) ) ) → ∃ 𝑏𝑈 ran 𝐹𝑏 )
25 17 24 rexlimddv ( 𝜑 → ∃ 𝑏𝑈 ran 𝐹𝑏 )
26 1 2 25 mnuss2d ( 𝜑 → ran 𝐹𝑈 )