Theorem seppsepf 47273

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 4037	. . . 4 ⊢ (𝑆 = (◡𝑓 “ {0}) → 𝑆 ⊆ (◡𝑓 “ {0}))
3		eqimss 4037	. . . 4 ⊢ (𝑇 = (◡𝑓 “ {1}) → 𝑇 ⊆ (◡𝑓 “ {1}))
4	2, 3	anim12i 613	. . 3 ⊢ ((𝑆 = (◡𝑓 “ {0}) ∧ 𝑇 = (◡𝑓 “ {1})) → (𝑆 ⊆ (◡𝑓 “ {0}) ∧ 𝑇 ⊆ (◡𝑓 “ {1})))
5	4	reximi 3084	. 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 396   = wceq 1541  ∃wrex 3070   ⊆ wss 3945  {csn 4623  ^◡ccnv 5669   “ cima 5673  (class class class)co 7394  0cc0 11094  1c1 11095   Cn ccn 22659  IIcii 24322

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476  df-in 3952  df-ss 3962

This theorem is referenced by: (None)