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Metamath Proof Explorer

Theorem frinfm

Description: A subset of a well-founded set has an infimum. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Assertion frinfm ( ( 𝑅 Fr 𝐴 ∧ ( 𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 fri ⊢ ( ( ( 𝐵 ∈ 𝐶 ∧ 𝑅 Fr 𝐴 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 )
2 1 ancom1s ⊢ ( ( ( 𝑅 Fr 𝐴 ∧ 𝐵 ∈ 𝐶 ) ∧ ( 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 )
3 2 exp43 ⊢ ( 𝑅 Fr 𝐴 → ( 𝐵 ∈ 𝐶 → ( 𝐵 ⊆ 𝐴 → ( 𝐵 ≠ ∅ → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) ) )
4 3 3imp2 ⊢ ( ( 𝑅 Fr 𝐴 ∧ ( 𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 )
5 ssel2 ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝑥 ∈ 𝐵 ) → 𝑥 ∈ 𝐴 )
6 5 adantrr ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) → 𝑥 ∈ 𝐴 )
7 vex ⊢ 𝑥 ∈ V
8 vex ⊢ 𝑦 ∈ V
9 7 8 brcnv ⊢ ( 𝑥 ◡ 𝑅 𝑦 ↔ 𝑦 𝑅 𝑥 )
10 9 biimpi ⊢ ( 𝑥 ◡ 𝑅 𝑦 → 𝑦 𝑅 𝑥 )
11 10 con3i ⊢ ( ¬ 𝑦 𝑅 𝑥 → ¬ 𝑥 ◡ 𝑅 𝑦 )
12 11 ralimi ⊢ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 → ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 )
13 12 ad2antll ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) → ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 )
14 breq2 ⊢ ( 𝑧 = 𝑥 → ( 𝑦 ◡ 𝑅 𝑧 ↔ 𝑦 ◡ 𝑅 𝑥 ) )
15 14 rspcev ⊢ ( ( 𝑥 ∈ 𝐵 ∧ 𝑦 ◡ 𝑅 𝑥 ) → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 )
16 15 ex ⊢ ( 𝑥 ∈ 𝐵 → ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) )
17 16 ralrimivw ⊢ ( 𝑥 ∈ 𝐵 → ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) )
18 17 ad2antrl ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) → ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) )
19 6 13 18 jca32 ⊢ ( ( 𝐵 ⊆ 𝐴 ∧ ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) ) → ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) ) )
20 19 ex ⊢ ( 𝐵 ⊆ 𝐴 → ( ( 𝑥 ∈ 𝐵 ∧ ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ) → ( 𝑥 ∈ 𝐴 ∧ ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) ) ) )
21 20 reximdv2 ⊢ ( 𝐵 ⊆ 𝐴 → ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) ) )
22 21 adantl ⊢ ( ( 𝑅 Fr 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) ) )
23 22 3ad2antr2 ⊢ ( ( 𝑅 Fr 𝐴 ∧ ( 𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ( ∃ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) ) )
24 4 23 mpd ⊢ ( ( 𝑅 Fr 𝐴 ∧ ( 𝐵 ∈ 𝐶 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅ ) ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 ◡ 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 ◡ 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 ◡ 𝑅 𝑧 ) ) )