Theorem seppsepf 47715

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 4032	. . . 4 ⊢ (𝑆 = (◡𝑓 “ {0}) → 𝑆 ⊆ (◡𝑓 “ {0}))
3		eqimss 4032	. . . 4 ⊢ (𝑇 = (◡𝑓 “ {1}) → 𝑇 ⊆ (◡𝑓 “ {1}))
4	2, 3	anim12i 612	. . 3 ⊢ ((𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})) → (𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
5	4	reximi 3076	. 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 17	1 ⊢ (𝜑 → ∃𝑓 ∈ (𝐽 Cn II)(𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533  ∃wrex 3062   ⊆ wss 3940  {csn 4620  ^◡ccnv 5665   “ cima 5669  (class class class)co 7401  0cc0 11105  1c1 11106   Cn ccn 23038  IIcii 24705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rex 3063  df-v 3468  df-in 3947  df-ss 3957

This theorem is referenced by: (None)