Theorem fun11uni

Description: The union of a chain (with respect to inclusion) of one-to-one functions is a one-to-one function. (Contributed by NM, 11-Aug-2004)

	Ref	Expression
Assertion	fun11uni	⊢ ( ∀ 𝑓 ∈ 𝐴 ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → ( Fun ∪ 𝐴 ∧ Fun ◡ ∪ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) → Fun 𝑓 )
2	1	anim1i	⊢ ( ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → ( Fun 𝑓 ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) )
3	2	ralimi	⊢ ( ∀ 𝑓 ∈ 𝐴 ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → ∀ 𝑓 ∈ 𝐴 ( Fun 𝑓 ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) )
4		fununi	⊢ ( ∀ 𝑓 ∈ 𝐴 ( Fun 𝑓 ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → Fun ∪ 𝐴 )
5	3 4	syl	⊢ ( ∀ 𝑓 ∈ 𝐴 ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → Fun ∪ 𝐴 )
6		simpr	⊢ ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) → Fun ◡ 𝑓 )
7	6	anim1i	⊢ ( ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → ( Fun ◡ 𝑓 ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) )
8	7	ralimi	⊢ ( ∀ 𝑓 ∈ 𝐴 ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → ∀ 𝑓 ∈ 𝐴 ( Fun ◡ 𝑓 ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) )
9		funcnvuni	⊢ ( ∀ 𝑓 ∈ 𝐴 ( Fun ◡ 𝑓 ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → Fun ◡ ∪ 𝐴 )
10	8 9	syl	⊢ ( ∀ 𝑓 ∈ 𝐴 ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → Fun ◡ ∪ 𝐴 )
11	5 10	jca	⊢ ( ∀ 𝑓 ∈ 𝐴 ( ( Fun 𝑓 ∧ Fun ◡ 𝑓 ) ∧ ∀ 𝑔 ∈ 𝐴 ( 𝑓 ⊆ 𝑔 ∨ 𝑔 ⊆ 𝑓 ) ) → ( Fun ∪ 𝐴 ∧ Fun ◡ ∪ 𝐴 ) )