Theorem tfsconcatrnss

Description: The concatenation of transfinite sequences yields elements from a class iff both sequences yield elements from that class. (Contributed by RP, 2-Mar-2025)

	Ref	Expression
Hypothesis	tfsconcat.op	⊢ + = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( dom 𝑎 +o dom 𝑏 ) ∖ dom 𝑎 ) ∧ ∃ 𝑧 ∈ dom 𝑏 ( 𝑥 = ( dom 𝑎 +o 𝑧 ) ∧ 𝑦 = ( 𝑏 ‘ 𝑧 ) ) ) } ) )
Assertion	tfsconcatrnss	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( ran ( 𝐴 + 𝐵 ) ⊆ 𝑋 ↔ ( ran 𝐴 ⊆ 𝑋 ∧ ran 𝐵 ⊆ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tfsconcat.op	⊢ + = ( 𝑎 ∈ V , 𝑏 ∈ V ↦ ( 𝑎 ∪ { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ ( ( dom 𝑎 +o dom 𝑏 ) ∖ dom 𝑎 ) ∧ ∃ 𝑧 ∈ dom 𝑏 ( 𝑥 = ( dom 𝑎 +o 𝑧 ) ∧ 𝑦 = ( 𝑏 ‘ 𝑧 ) ) ) } ) )
2	1	tfsconcatrn	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ran ( 𝐴 + 𝐵 ) = ( ran 𝐴 ∪ ran 𝐵 ) )
3	2	sseq1d	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( ran ( 𝐴 + 𝐵 ) ⊆ 𝑋 ↔ ( ran 𝐴 ∪ ran 𝐵 ) ⊆ 𝑋 ) )
4		unss	⊢ ( ( ran 𝐴 ⊆ 𝑋 ∧ ran 𝐵 ⊆ 𝑋 ) ↔ ( ran 𝐴 ∪ ran 𝐵 ) ⊆ 𝑋 )
5	3 4	bitr4di	⊢ ( ( ( 𝐴 Fn 𝐶 ∧ 𝐵 Fn 𝐷 ) ∧ ( 𝐶 ∈ On ∧ 𝐷 ∈ On ) ) → ( ran ( 𝐴 + 𝐵 ) ⊆ 𝑋 ↔ ( ran 𝐴 ⊆ 𝑋 ∧ ran 𝐵 ⊆ 𝑋 ) ) )