Theorem seppsepf 49857

Description: If two sets are precisely separated by a continuous function, then they are separated by the continuous function. (Contributed by Zhi Wang, 9-Sep-2024.)

Hypothesis

Ref	Expression
seppsepf.1	⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))

Assertion

Ref	Expression
seppsepf	⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))

Proof of Theorem seppsepf

Step	Hyp	Ref	Expression
1		seppsepf.1	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})))
2		eqimss 3992	. . . 4 ⊢ (𝑆 = (◡𝑓 “ {0}) → 𝑆 ⊆ (◡𝑓 “ {0}))
3		eqimss 3992	. . . 4 ⊢ (𝑇 = (◡𝑓 “ {1}) → 𝑇 ⊆ (◡𝑓 “ {1}))
4	2, 3	anim12i 625	. . 3 ⊢ ((𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})) → (𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
5	4	reximi 3102	. 2 ⊢ (∃𝑓 ∈ (𝐽 Cn II)(𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})) → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
6	1, 5	syl 18	1 ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wrex 3088   ⊆ wss 3902  {csn 4587  ^◡ccnv 5658   “ cima 5662  (class class class)co 7416  0cc0 11127  1c1 11128   Cn ccn 23453  IIcii 25107

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2754  df-rex 3089  df-ss 3919

This theorem is used by: (None)