Metamath Proof Explorer


Theorem zorn2lem5

Description: Lemma for zorn2 . (Contributed by NM, 4-Apr-1997) (Revised by Mario Carneiro, 9-May-2015)

Ref Expression
Hypotheses zorn2lem.3 𝐹 = recs ( ( 𝑓 ∈ V ↦ ( 𝑣𝐶𝑢𝐶 ¬ 𝑢 𝑤 𝑣 ) ) )
zorn2lem.4 𝐶 = { 𝑧𝐴 ∣ ∀ 𝑔 ∈ ran 𝑓 𝑔 𝑅 𝑧 }
zorn2lem.5 𝐷 = { 𝑧𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹𝑥 ) 𝑔 𝑅 𝑧 }
zorn2lem.7 𝐻 = { 𝑧𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹𝑦 ) 𝑔 𝑅 𝑧 }
Assertion zorn2lem5 ( ( ( 𝑤 We 𝐴𝑥 ∈ On ) ∧ ∀ 𝑦𝑥 𝐻 ≠ ∅ ) → ( 𝐹𝑥 ) ⊆ 𝐴 )

Proof

Step Hyp Ref Expression
1 zorn2lem.3 𝐹 = recs ( ( 𝑓 ∈ V ↦ ( 𝑣𝐶𝑢𝐶 ¬ 𝑢 𝑤 𝑣 ) ) )
2 zorn2lem.4 𝐶 = { 𝑧𝐴 ∣ ∀ 𝑔 ∈ ran 𝑓 𝑔 𝑅 𝑧 }
3 zorn2lem.5 𝐷 = { 𝑧𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹𝑥 ) 𝑔 𝑅 𝑧 }
4 zorn2lem.7 𝐻 = { 𝑧𝐴 ∣ ∀ 𝑔 ∈ ( 𝐹𝑦 ) 𝑔 𝑅 𝑧 }
5 1 tfr1 𝐹 Fn On
6 fnfun ( 𝐹 Fn On → Fun 𝐹 )
7 5 6 ax-mp Fun 𝐹
8 fvelima ( ( Fun 𝐹𝑠 ∈ ( 𝐹𝑥 ) ) → ∃ 𝑦𝑥 ( 𝐹𝑦 ) = 𝑠 )
9 7 8 mpan ( 𝑠 ∈ ( 𝐹𝑥 ) → ∃ 𝑦𝑥 ( 𝐹𝑦 ) = 𝑠 )
10 nfv 𝑦 ( 𝑤 We 𝐴𝑥 ∈ On )
11 nfra1 𝑦𝑦𝑥 𝐻 ≠ ∅
12 10 11 nfan 𝑦 ( ( 𝑤 We 𝐴𝑥 ∈ On ) ∧ ∀ 𝑦𝑥 𝐻 ≠ ∅ )
13 nfv 𝑦 𝑠𝐴
14 df-ral ( ∀ 𝑦𝑥 𝐻 ≠ ∅ ↔ ∀ 𝑦 ( 𝑦𝑥𝐻 ≠ ∅ ) )
15 onelon ( ( 𝑥 ∈ On ∧ 𝑦𝑥 ) → 𝑦 ∈ On )
16 4 ssrab3 𝐻𝐴
17 1 2 4 zorn2lem1 ( ( 𝑦 ∈ On ∧ ( 𝑤 We 𝐴𝐻 ≠ ∅ ) ) → ( 𝐹𝑦 ) ∈ 𝐻 )
18 16 17 sselid ( ( 𝑦 ∈ On ∧ ( 𝑤 We 𝐴𝐻 ≠ ∅ ) ) → ( 𝐹𝑦 ) ∈ 𝐴 )
19 eleq1 ( ( 𝐹𝑦 ) = 𝑠 → ( ( 𝐹𝑦 ) ∈ 𝐴𝑠𝐴 ) )
20 18 19 syl5ib ( ( 𝐹𝑦 ) = 𝑠 → ( ( 𝑦 ∈ On ∧ ( 𝑤 We 𝐴𝐻 ≠ ∅ ) ) → 𝑠𝐴 ) )
21 15 20 sylani ( ( 𝐹𝑦 ) = 𝑠 → ( ( ( 𝑥 ∈ On ∧ 𝑦𝑥 ) ∧ ( 𝑤 We 𝐴𝐻 ≠ ∅ ) ) → 𝑠𝐴 ) )
22 21 com12 ( ( ( 𝑥 ∈ On ∧ 𝑦𝑥 ) ∧ ( 𝑤 We 𝐴𝐻 ≠ ∅ ) ) → ( ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) )
23 22 exp43 ( 𝑥 ∈ On → ( 𝑦𝑥 → ( 𝑤 We 𝐴 → ( 𝐻 ≠ ∅ → ( ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) ) ) ) )
24 23 com3r ( 𝑤 We 𝐴 → ( 𝑥 ∈ On → ( 𝑦𝑥 → ( 𝐻 ≠ ∅ → ( ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) ) ) ) )
25 24 imp ( ( 𝑤 We 𝐴𝑥 ∈ On ) → ( 𝑦𝑥 → ( 𝐻 ≠ ∅ → ( ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) ) ) )
26 25 a2d ( ( 𝑤 We 𝐴𝑥 ∈ On ) → ( ( 𝑦𝑥𝐻 ≠ ∅ ) → ( 𝑦𝑥 → ( ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) ) ) )
27 26 spsd ( ( 𝑤 We 𝐴𝑥 ∈ On ) → ( ∀ 𝑦 ( 𝑦𝑥𝐻 ≠ ∅ ) → ( 𝑦𝑥 → ( ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) ) ) )
28 14 27 syl5bi ( ( 𝑤 We 𝐴𝑥 ∈ On ) → ( ∀ 𝑦𝑥 𝐻 ≠ ∅ → ( 𝑦𝑥 → ( ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) ) ) )
29 28 imp ( ( ( 𝑤 We 𝐴𝑥 ∈ On ) ∧ ∀ 𝑦𝑥 𝐻 ≠ ∅ ) → ( 𝑦𝑥 → ( ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) ) )
30 12 13 29 rexlimd ( ( ( 𝑤 We 𝐴𝑥 ∈ On ) ∧ ∀ 𝑦𝑥 𝐻 ≠ ∅ ) → ( ∃ 𝑦𝑥 ( 𝐹𝑦 ) = 𝑠𝑠𝐴 ) )
31 9 30 syl5 ( ( ( 𝑤 We 𝐴𝑥 ∈ On ) ∧ ∀ 𝑦𝑥 𝐻 ≠ ∅ ) → ( 𝑠 ∈ ( 𝐹𝑥 ) → 𝑠𝐴 ) )
32 31 ssrdv ( ( ( 𝑤 We 𝐴𝑥 ∈ On ) ∧ ∀ 𝑦𝑥 𝐻 ≠ ∅ ) → ( 𝐹𝑥 ) ⊆ 𝐴 )