Metamath Proof Explorer


Theorem cardinfima

Description: If a mapping to cardinals has an infinite value, then the union of its image is an infinite cardinal. Corollary 11.17 of TakeutiZaring p. 104. (Contributed by NM, 4-Nov-2004)

Ref Expression
Assertion cardinfima ( 𝐴𝐵 → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ∃ 𝑥𝐴 ( 𝐹𝑥 ) ∈ ran ℵ ) → ( 𝐹𝐴 ) ∈ ran ℵ ) )

Proof

Step Hyp Ref Expression
1 elex ( 𝐴𝐵𝐴 ∈ V )
2 isinfcard ( ( ω ⊆ ( 𝐹𝑥 ) ∧ ( card ‘ ( 𝐹𝑥 ) ) = ( 𝐹𝑥 ) ) ↔ ( 𝐹𝑥 ) ∈ ran ℵ )
3 2 bicomi ( ( 𝐹𝑥 ) ∈ ran ℵ ↔ ( ω ⊆ ( 𝐹𝑥 ) ∧ ( card ‘ ( 𝐹𝑥 ) ) = ( 𝐹𝑥 ) ) )
4 3 simplbi ( ( 𝐹𝑥 ) ∈ ran ℵ → ω ⊆ ( 𝐹𝑥 ) )
5 ffn ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) → 𝐹 Fn 𝐴 )
6 fnfvelrn ( ( 𝐹 Fn 𝐴𝑥𝐴 ) → ( 𝐹𝑥 ) ∈ ran 𝐹 )
7 6 ex ( 𝐹 Fn 𝐴 → ( 𝑥𝐴 → ( 𝐹𝑥 ) ∈ ran 𝐹 ) )
8 fnima ( 𝐹 Fn 𝐴 → ( 𝐹𝐴 ) = ran 𝐹 )
9 8 eleq2d ( 𝐹 Fn 𝐴 → ( ( 𝐹𝑥 ) ∈ ( 𝐹𝐴 ) ↔ ( 𝐹𝑥 ) ∈ ran 𝐹 ) )
10 7 9 sylibrd ( 𝐹 Fn 𝐴 → ( 𝑥𝐴 → ( 𝐹𝑥 ) ∈ ( 𝐹𝐴 ) ) )
11 elssuni ( ( 𝐹𝑥 ) ∈ ( 𝐹𝐴 ) → ( 𝐹𝑥 ) ⊆ ( 𝐹𝐴 ) )
12 10 11 syl6 ( 𝐹 Fn 𝐴 → ( 𝑥𝐴 → ( 𝐹𝑥 ) ⊆ ( 𝐹𝐴 ) ) )
13 12 imp ( ( 𝐹 Fn 𝐴𝑥𝐴 ) → ( 𝐹𝑥 ) ⊆ ( 𝐹𝐴 ) )
14 5 13 sylan ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ 𝑥𝐴 ) → ( 𝐹𝑥 ) ⊆ ( 𝐹𝐴 ) )
15 4 14 sylan9ssr ( ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ 𝑥𝐴 ) ∧ ( 𝐹𝑥 ) ∈ ran ℵ ) → ω ⊆ ( 𝐹𝐴 ) )
16 15 anasss ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥𝐴 ∧ ( 𝐹𝑥 ) ∈ ran ℵ ) ) → ω ⊆ ( 𝐹𝐴 ) )
17 16 a1i ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥𝐴 ∧ ( 𝐹𝑥 ) ∈ ran ℵ ) ) → ω ⊆ ( 𝐹𝐴 ) ) )
18 carduniima ( 𝐴 ∈ V → ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) → ( 𝐹𝐴 ) ∈ ( ω ∪ ran ℵ ) ) )
19 iscard3 ( ( card ‘ ( 𝐹𝐴 ) ) = ( 𝐹𝐴 ) ↔ ( 𝐹𝐴 ) ∈ ( ω ∪ ran ℵ ) )
20 18 19 syl6ibr ( 𝐴 ∈ V → ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) → ( card ‘ ( 𝐹𝐴 ) ) = ( 𝐹𝐴 ) ) )
21 20 adantrd ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥𝐴 ∧ ( 𝐹𝑥 ) ∈ ran ℵ ) ) → ( card ‘ ( 𝐹𝐴 ) ) = ( 𝐹𝐴 ) ) )
22 17 21 jcad ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥𝐴 ∧ ( 𝐹𝑥 ) ∈ ran ℵ ) ) → ( ω ⊆ ( 𝐹𝐴 ) ∧ ( card ‘ ( 𝐹𝐴 ) ) = ( 𝐹𝐴 ) ) ) )
23 isinfcard ( ( ω ⊆ ( 𝐹𝐴 ) ∧ ( card ‘ ( 𝐹𝐴 ) ) = ( 𝐹𝐴 ) ) ↔ ( 𝐹𝐴 ) ∈ ran ℵ )
24 22 23 syl6ib ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ( 𝑥𝐴 ∧ ( 𝐹𝑥 ) ∈ ran ℵ ) ) → ( 𝐹𝐴 ) ∈ ran ℵ ) )
25 24 exp4d ( 𝐴 ∈ V → ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) → ( 𝑥𝐴 → ( ( 𝐹𝑥 ) ∈ ran ℵ → ( 𝐹𝐴 ) ∈ ran ℵ ) ) ) )
26 25 imp ( ( 𝐴 ∈ V ∧ 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ) → ( 𝑥𝐴 → ( ( 𝐹𝑥 ) ∈ ran ℵ → ( 𝐹𝐴 ) ∈ ran ℵ ) ) )
27 26 rexlimdv ( ( 𝐴 ∈ V ∧ 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ) → ( ∃ 𝑥𝐴 ( 𝐹𝑥 ) ∈ ran ℵ → ( 𝐹𝐴 ) ∈ ran ℵ ) )
28 27 expimpd ( 𝐴 ∈ V → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ∃ 𝑥𝐴 ( 𝐹𝑥 ) ∈ ran ℵ ) → ( 𝐹𝐴 ) ∈ ran ℵ ) )
29 1 28 syl ( 𝐴𝐵 → ( ( 𝐹 : 𝐴 ⟶ ( ω ∪ ran ℵ ) ∧ ∃ 𝑥𝐴 ( 𝐹𝑥 ) ∈ ran ℵ ) → ( 𝐹𝐴 ) ∈ ran ℵ ) )