Metamath Proof Explorer

Theorem weiunlem

Description: Lemma for weiunpo , weiunso , weiunfr , and weiunse . (Contributed by Matthew House, 23-Aug-2025)

Ref Expression
Hypotheses weiun.1 ⊢ 𝐹 = ( 𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) )
weiun.2 ⊢ 𝑇 = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐹 ‘ 𝑧 ) ∨ ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ∧ 𝑦 ⦋ ( 𝐹 ‘ 𝑦 ) / 𝑥 ⦌ 𝑆 𝑧 ) ) ) }
weiunlem.3 ⊢ ( 𝜑 → 𝑅 We 𝐴 )
weiunlem.4 ⊢ ( 𝜑 → 𝑅 Se 𝐴 )
Assertion weiunlem ( 𝜑 → ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ∧ ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) )

Proof

Step Hyp Ref Expression
1 weiun.1 ⊢ 𝐹 = ( 𝑤 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↦ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) )
2 weiun.2 ⊢ 𝑇 = { ⟨ 𝑦 , 𝑧 ⟩ ∣ ( ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) ∧ ( ( 𝐹 ‘ 𝑦 ) 𝑅 ( 𝐹 ‘ 𝑧 ) ∨ ( ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) ∧ 𝑦 ⦋ ( 𝐹 ‘ 𝑦 ) / 𝑥 ⦌ 𝑆 𝑧 ) ) ) }
3 weiunlem.3 ⊢ ( 𝜑 → 𝑅 We 𝐴 )
4 weiunlem.4 ⊢ ( 𝜑 → 𝑅 Se 𝐴 )
5 riotaex ⊢ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) ∈ V
6 5 1 fnmpti ⊢ 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵
7 6 a1i ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵 )
8 breq2 ⊢ ( 𝑢 = 𝑟 → ( 𝑣 𝑅 𝑢 ↔ 𝑣 𝑅 𝑟 ) )
9 8 notbid ⊢ ( 𝑢 = 𝑟 → ( ¬ 𝑣 𝑅 𝑢 ↔ ¬ 𝑣 𝑅 𝑟 ) )
10 9 ralbidv ⊢ ( 𝑢 = 𝑟 → ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ↔ ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ) )
11 10 cbvriotavw ⊢ ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 )
12 eleq1w ⊢ ( 𝑤 = 𝑡 → ( 𝑤 ∈ 𝐵 ↔ 𝑡 ∈ 𝐵 ) )
13 12 rabbidv ⊢ ( 𝑤 = 𝑡 → { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } = { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } )
14 breq1 ⊢ ( 𝑣 = 𝑠 → ( 𝑣 𝑅 𝑟 ↔ 𝑠 𝑅 𝑟 ) )
15 14 notbid ⊢ ( 𝑣 = 𝑠 → ( ¬ 𝑣 𝑅 𝑟 ↔ ¬ 𝑠 𝑅 𝑟 ) )
16 15 cbvralvw ⊢ ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ↔ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 )
17 13 raleqdv ⊢ ( 𝑤 = 𝑡 → ( ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ↔ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) )
18 16 17 bitrid ⊢ ( 𝑤 = 𝑡 → ( ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ↔ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) )
19 13 18 riotaeqbidv ⊢ ( 𝑤 = 𝑡 → ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑟 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) )
20 11 19 eqtrid ⊢ ( 𝑤 = 𝑡 → ( ℩ 𝑢 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ∀ 𝑣 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑤 ∈ 𝐵 } ¬ 𝑣 𝑅 𝑢 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) )
21 20 1 5 fvmpt3i ⊢ ( 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) )
22 21 adantl ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ( 𝐹 ‘ 𝑡 ) = ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) )
23 eliun ⊢ ( 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 )
24 rabn0 ⊢ ( { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ ↔ ∃ 𝑥 ∈ 𝐴 𝑡 ∈ 𝐵 )
25 23 24 bitr4i ⊢ ( 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ )
26 ssrab2 ⊢ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ⊆ 𝐴
27 wereu2 ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ⊆ 𝐴 ∧ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ ) ) → ∃! 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 )
28 26 27 mpanr1 ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ ) → ∃! 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 )
29 25 28 sylan2b ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ∃! 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 )
30 riotacl2 ⊢ ( ∃! 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 → ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } )
31 29 30 syl ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ( ℩ 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } )
32 22 31 eqeltrd ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ( 𝐹 ‘ 𝑡 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } )
33 elrabi ⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } → ( 𝐹 ‘ 𝑡 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } )
34 elrabi ⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } → ( 𝐹 ‘ 𝑡 ) ∈ 𝐴 )
35 32 33 34 3syl ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ( 𝐹 ‘ 𝑡 ) ∈ 𝐴 )
36 35 ralrimiva ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝐹 ‘ 𝑡 ) ∈ 𝐴 )
37 ffnfv ⊢ ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ↔ ( 𝐹 Fn ∪ 𝑥 ∈ 𝐴 𝐵 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ( 𝐹 ‘ 𝑡 ) ∈ 𝐴 ) )
38 7 36 37 sylanbrc ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 )
39 dfsbcq ⊢ ( 𝑠 = ( 𝐹 ‘ 𝑡 ) → ( [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ↔ [ ( 𝐹 ‘ 𝑡 ) / 𝑥 ] 𝑡 ∈ 𝐵 ) )
40 nfcv ⊢ Ⅎ 𝑥 𝐴
41 40 elrabsf ⊢ ( 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ↔ ( 𝑠 ∈ 𝐴 ∧ [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ) )
42 41 simprbi ⊢ ( 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } → [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 )
43 39 42 vtoclga ⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } → [ ( 𝐹 ‘ 𝑡 ) / 𝑥 ] 𝑡 ∈ 𝐵 )
44 32 33 43 3syl ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → [ ( 𝐹 ‘ 𝑡 ) / 𝑥 ] 𝑡 ∈ 𝐵 )
45 sbcel2 ⊢ ( [ ( 𝐹 ‘ 𝑡 ) / 𝑥 ] 𝑡 ∈ 𝐵 ↔ 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 )
46 44 45 sylib ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 )
47 46 ralrimiva ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 )
48 sbcel2 ⊢ ( [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ↔ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 )
49 48 anbi2i ⊢ ( ( 𝑠 ∈ 𝐴 ∧ [ 𝑠 / 𝑥 ] 𝑡 ∈ 𝐵 ) ↔ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) )
50 41 49 bitri ⊢ ( 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ↔ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) )
51 50 bilanri ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } )
52 51 ne0d ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ≠ ∅ )
53 52 25 sylibr ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 )
54 breq2 ⊢ ( 𝑟 = ( 𝐹 ‘ 𝑡 ) → ( 𝑠 𝑅 𝑟 ↔ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) )
55 54 notbid ⊢ ( 𝑟 = ( 𝐹 ‘ 𝑡 ) → ( ¬ 𝑠 𝑅 𝑟 ↔ ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) )
56 55 ralbidv ⊢ ( 𝑟 = ( 𝐹 ‘ 𝑡 ) → ( ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 ↔ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) )
57 56 elrab ⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } ↔ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∧ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) )
58 57 simprbi ⊢ ( ( 𝐹 ‘ 𝑡 ) ∈ { 𝑟 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ∣ ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 𝑟 } → ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) )
59 32 58 syl ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) → ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) )
60 53 59 syldan ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) )
61 rsp ⊢ ( ∀ 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) → ( 𝑠 ∈ { 𝑥 ∈ 𝐴 ∣ 𝑡 ∈ 𝐵 } → ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) )
62 60 51 61 sylc ⊢ ( ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ) ) → ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) )
63 62 ralrimivva ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) )
64 38 47 63 3jca ⊢ ( ( 𝑅 We 𝐴 ∧ 𝑅 Se 𝐴 ) → ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ∧ ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) )
65 3 4 64 syl2anc ⊢ ( 𝜑 → ( 𝐹 : ∪ 𝑥 ∈ 𝐴 𝐵 ⟶ 𝐴 ∧ ∀ 𝑡 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 𝑡 ∈ ⦋ ( 𝐹 ‘ 𝑡 ) / 𝑥 ⦌ 𝐵 ∧ ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ ⦋ 𝑠 / 𝑥 ⦌ 𝐵 ¬ 𝑠 𝑅 ( 𝐹 ‘ 𝑡 ) ) )