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Theorem ucncn 22870
 Description: Uniform continuity implies continuity. Deduction form. Proposition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 30-Nov-2017.)
Hypotheses
Ref Expression
ucncn.j 𝐽 = (TopOpen‘𝑅)
ucncn.k 𝐾 = (TopOpen‘𝑆)
ucncn.1 (𝜑𝑅 ∈ UnifSp)
ucncn.2 (𝜑𝑆 ∈ UnifSp)
ucncn.3 (𝜑𝑅 ∈ TopSp)
ucncn.4 (𝜑𝑆 ∈ TopSp)
ucncn.5 (𝜑𝐹 ∈ ((UnifSt‘𝑅) Cnu(UnifSt‘𝑆)))
Assertion
Ref Expression
ucncn (𝜑𝐹 ∈ (𝐽 Cn 𝐾))

Proof of Theorem ucncn
Dummy variables 𝑟 𝑎 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ucncn.5 . . . 4 (𝜑𝐹 ∈ ((UnifSt‘𝑅) Cnu(UnifSt‘𝑆)))
2 ucncn.1 . . . . . 6 (𝜑𝑅 ∈ UnifSp)
3 eqid 2820 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
4 eqid 2820 . . . . . . . 8 (UnifSt‘𝑅) = (UnifSt‘𝑅)
5 ucncn.j . . . . . . . 8 𝐽 = (TopOpen‘𝑅)
63, 4, 5isusp 22846 . . . . . . 7 (𝑅 ∈ UnifSp ↔ ((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑅))))
76simplbi 500 . . . . . 6 (𝑅 ∈ UnifSp → (UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)))
82, 7syl 17 . . . . 5 (𝜑 → (UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)))
9 ucncn.2 . . . . . 6 (𝜑𝑆 ∈ UnifSp)
10 eqid 2820 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
11 eqid 2820 . . . . . . . 8 (UnifSt‘𝑆) = (UnifSt‘𝑆)
12 ucncn.k . . . . . . . 8 𝐾 = (TopOpen‘𝑆)
1310, 11, 12isusp 22846 . . . . . . 7 (𝑆 ∈ UnifSp ↔ ((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) ∧ 𝐾 = (unifTop‘(UnifSt‘𝑆))))
1413simplbi 500 . . . . . 6 (𝑆 ∈ UnifSp → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
159, 14syl 17 . . . . 5 (𝜑 → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
16 isucn 22863 . . . . 5 (((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆))) → (𝐹 ∈ ((UnifSt‘𝑅) Cnu(UnifSt‘𝑆)) ↔ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))))
178, 15, 16syl2anc 586 . . . 4 (𝜑 → (𝐹 ∈ ((UnifSt‘𝑅) Cnu(UnifSt‘𝑆)) ↔ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))))
181, 17mpbid 234 . . 3 (𝜑 → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))))
1918simpld 497 . 2 (𝜑𝐹:(Base‘𝑅)⟶(Base‘𝑆))
20 cnvimass 5925 . . . . 5 (𝐹𝑎) ⊆ dom 𝐹
2119fdmd 6499 . . . . . 6 (𝜑 → dom 𝐹 = (Base‘𝑅))
2221adantr 483 . . . . 5 ((𝜑𝑎𝐾) → dom 𝐹 = (Base‘𝑅))
2320, 22sseqtrid 3998 . . . 4 ((𝜑𝑎𝐾) → (𝐹𝑎) ⊆ (Base‘𝑅))
24 simplll 773 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝜑)
25 simpr 487 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑠 ∈ (UnifSt‘𝑆))
2623ad2antrr 724 . . . . . . . . . 10 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → (𝐹𝑎) ⊆ (Base‘𝑅))
27 simplr 767 . . . . . . . . . 10 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑥 ∈ (𝐹𝑎))
2826, 27sseldd 3947 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
2918simprd 498 . . . . . . . . . . . 12 (𝜑 → ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3029r19.21bi 3195 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (UnifSt‘𝑆)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
31 r19.12 3311 . . . . . . . . . . 11 (∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) → ∀𝑥 ∈ (Base‘𝑅)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3230, 31syl 17 . . . . . . . . . 10 ((𝜑𝑠 ∈ (UnifSt‘𝑆)) → ∀𝑥 ∈ (Base‘𝑅)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3332r19.21bi 3195 . . . . . . . . 9 (((𝜑𝑠 ∈ (UnifSt‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3424, 25, 28, 33syl21anc 835 . . . . . . . 8 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3534adantr 483 . . . . . . 7 (((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3624ad3antrrr 728 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝜑)
378ad5antr 732 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)))
38 simpr 487 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → 𝑟 ∈ (UnifSt‘𝑅))
39 ustrel 22796 . . . . . . . . . . . 12 (((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑟)
4037, 38, 39syl2anc 586 . . . . . . . . . . 11 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑟)
4140adantr 483 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → Rel 𝑟)
4236, 8syl 17 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → (UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)))
43 simplr 767 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝑟 ∈ (UnifSt‘𝑅))
4428ad3antrrr 728 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝑥 ∈ (Base‘𝑅))
45 ustimasn 22813 . . . . . . . . . . 11 (((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝑟 ∈ (UnifSt‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
4642, 43, 44, 45syl3anc 1367 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
47 simpr 487 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
48 simplr 767 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → 𝑧 ∈ (Base‘𝑅))
49 simpllr 774 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎)
5015ad5antr 732 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
51 simpllr 774 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → 𝑠 ∈ (UnifSt‘𝑆))
52 ustrel 22796 . . . . . . . . . . . . . . . . . . . 20 (((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → Rel 𝑠)
5350, 51, 52syl2anc 586 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑠)
54 elrelimasn 5929 . . . . . . . . . . . . . . . . . . 19 (Rel 𝑠 → ((𝐹𝑧) ∈ (𝑠 “ {(𝐹𝑥)}) ↔ (𝐹𝑥)𝑠(𝐹𝑧)))
5553, 54syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → ((𝐹𝑧) ∈ (𝑠 “ {(𝐹𝑥)}) ↔ (𝐹𝑥)𝑠(𝐹𝑧)))
5655biimpar 480 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ (𝑠 “ {(𝐹𝑥)}))
5749, 56sseldd 3947 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ 𝑎)
5857adantlr 713 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ 𝑎)
59 ffn 6490 . . . . . . . . . . . . . . . . 17 (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → 𝐹 Fn (Base‘𝑅))
60 elpreima 6804 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (Base‘𝑅) → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧 ∈ (Base‘𝑅) ∧ (𝐹𝑧) ∈ 𝑎)))
6119, 59, 603syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧 ∈ (Base‘𝑅) ∧ (𝐹𝑧) ∈ 𝑎)))
6261ad7antr 736 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧 ∈ (Base‘𝑅) ∧ (𝐹𝑧) ∈ 𝑎)))
6348, 58, 62mpbir2and 711 . . . . . . . . . . . . . 14 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → 𝑧 ∈ (𝐹𝑎))
6463ex 415 . . . . . . . . . . . . 13 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) → ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
6564ralrimiva 3169 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
6665adantr 483 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
67 r19.26 3157 . . . . . . . . . . . 12 (∀𝑧 ∈ (Base‘𝑅)((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) ↔ (∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))))
68 pm3.33 763 . . . . . . . . . . . . 13 (((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → (𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
6968ralimi 3147 . . . . . . . . . . . 12 (∀𝑧 ∈ (Base‘𝑅)((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
7067, 69sylbir 237 . . . . . . . . . . 11 ((∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
7147, 66, 70syl2anc 586 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
72 simpl2l 1222 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → Rel 𝑟)
73 simpr 487 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (𝑟 “ {𝑥}))
74 elrelimasn 5929 . . . . . . . . . . . . . . 15 (Rel 𝑟 → (𝑦 ∈ (𝑟 “ {𝑥}) ↔ 𝑥𝑟𝑦))
7574biimpa 479 . . . . . . . . . . . . . 14 ((Rel 𝑟𝑦 ∈ (𝑟 “ {𝑥})) → 𝑥𝑟𝑦)
7672, 73, 75syl2anc 586 . . . . . . . . . . . . 13 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑥𝑟𝑦)
77 breq2 5046 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑥𝑟𝑧𝑥𝑟𝑦))
78 eleq1w 2893 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑧 ∈ (𝐹𝑎) ↔ 𝑦 ∈ (𝐹𝑎)))
7977, 78imbi12d 347 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)) ↔ (𝑥𝑟𝑦𝑦 ∈ (𝐹𝑎))))
80 simpl3 1189 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
81 simpl2r 1223 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
8281, 73sseldd 3947 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (Base‘𝑅))
8379, 80, 82rspcdva 3604 . . . . . . . . . . . . 13 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → (𝑥𝑟𝑦𝑦 ∈ (𝐹𝑎)))
8476, 83mpd 15 . . . . . . . . . . . 12 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (𝐹𝑎))
8584ex 415 . . . . . . . . . . 11 ((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) → (𝑦 ∈ (𝑟 “ {𝑥}) → 𝑦 ∈ (𝐹𝑎)))
8685ssrdv 3952 . . . . . . . . . 10 ((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
8736, 41, 46, 71, 86syl121anc 1371 . . . . . . . . 9 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
8887ex 415 . . . . . . . 8 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎)))
8988reximdva 3261 . . . . . . 7 (((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) → (∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) → ∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎)))
9035, 89mpd 15 . . . . . 6 (((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) → ∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
91 sneq 4553 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → {𝑦} = {(𝐹𝑥)})
9291imaeq2d 5905 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → (𝑠 “ {𝑦}) = (𝑠 “ {(𝐹𝑥)}))
9392sseq1d 3977 . . . . . . . 8 (𝑦 = (𝐹𝑥) → ((𝑠 “ {𝑦}) ⊆ 𝑎 ↔ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
9493rexbidv 3284 . . . . . . 7 (𝑦 = (𝐹𝑥) → (∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎 ↔ ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
95 simpr 487 . . . . . . . . . . 11 ((𝜑𝑎𝐾) → 𝑎𝐾)
9613simprbi 499 . . . . . . . . . . . . 13 (𝑆 ∈ UnifSp → 𝐾 = (unifTop‘(UnifSt‘𝑆)))
979, 96syl 17 . . . . . . . . . . . 12 (𝜑𝐾 = (unifTop‘(UnifSt‘𝑆)))
9897adantr 483 . . . . . . . . . . 11 ((𝜑𝑎𝐾) → 𝐾 = (unifTop‘(UnifSt‘𝑆)))
9995, 98eleqtrd 2913 . . . . . . . . . 10 ((𝜑𝑎𝐾) → 𝑎 ∈ (unifTop‘(UnifSt‘𝑆)))
100 elutop 22818 . . . . . . . . . . . 12 ((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) → (𝑎 ∈ (unifTop‘(UnifSt‘𝑆)) ↔ (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)))
10115, 100syl 17 . . . . . . . . . . 11 (𝜑 → (𝑎 ∈ (unifTop‘(UnifSt‘𝑆)) ↔ (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)))
102101adantr 483 . . . . . . . . . 10 ((𝜑𝑎𝐾) → (𝑎 ∈ (unifTop‘(UnifSt‘𝑆)) ↔ (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)))
10399, 102mpbid 234 . . . . . . . . 9 ((𝜑𝑎𝐾) → (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎))
104103simprd 498 . . . . . . . 8 ((𝜑𝑎𝐾) → ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)
105104adantr 483 . . . . . . 7 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)
106 elpreima 6804 . . . . . . . . . . 11 (𝐹 Fn (Base‘𝑅) → (𝑥 ∈ (𝐹𝑎) ↔ (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎)))
10719, 59, 1063syl 18 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐹𝑎) ↔ (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎)))
108107adantr 483 . . . . . . . . 9 ((𝜑𝑎𝐾) → (𝑥 ∈ (𝐹𝑎) ↔ (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎)))
109108biimpa 479 . . . . . . . 8 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎))
110109simprd 498 . . . . . . 7 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → (𝐹𝑥) ∈ 𝑎)
11194, 105, 110rspcdva 3604 . . . . . 6 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎)
11290, 111r19.29a 3276 . . . . 5 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
113112ralrimiva 3169 . . . 4 ((𝜑𝑎𝐾) → ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
1146simprbi 499 . . . . . . . 8 (𝑅 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑅)))
1152, 114syl 17 . . . . . . 7 (𝜑𝐽 = (unifTop‘(UnifSt‘𝑅)))
116115adantr 483 . . . . . 6 ((𝜑𝑎𝐾) → 𝐽 = (unifTop‘(UnifSt‘𝑅)))
117116eleq2d 2896 . . . . 5 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ 𝐽 ↔ (𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅))))
118 elutop 22818 . . . . . . 7 ((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) → ((𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅)) ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
1198, 118syl 17 . . . . . 6 (𝜑 → ((𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅)) ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
120119adantr 483 . . . . 5 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅)) ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
121117, 120bitrd 281 . . . 4 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ 𝐽 ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
12223, 113, 121mpbir2and 711 . . 3 ((𝜑𝑎𝐾) → (𝐹𝑎) ∈ 𝐽)
123122ralrimiva 3169 . 2 (𝜑 → ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)
124 ucncn.3 . . . 4 (𝜑𝑅 ∈ TopSp)
1253, 5istps 21518 . . . 4 (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑅)))
126124, 125sylib 220 . . 3 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑅)))
127 ucncn.4 . . . 4 (𝜑𝑆 ∈ TopSp)
12810, 12istps 21518 . . . 4 (𝑆 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝑆)))
129127, 128sylib 220 . . 3 (𝜑𝐾 ∈ (TopOn‘(Base‘𝑆)))
130 iscn 21819 . . 3 ((𝐽 ∈ (TopOn‘(Base‘𝑅)) ∧ 𝐾 ∈ (TopOn‘(Base‘𝑆))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)))
131126, 129, 130syl2anc 586 . 2 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)))
13219, 123, 131mpbir2and 711 1 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3125  ∃wrex 3126   ⊆ wss 3913  {csn 4543   class class class wbr 5042  ◡ccnv 5530  dom cdm 5531   “ cima 5534  Rel wrel 5536   Fn wfn 6326  ⟶wf 6327  ‘cfv 6331  (class class class)co 7133  Basecbs 16462  TopOpenctopn 16674  TopOnctopon 21494  TopSpctps 21516   Cn ccn 21808  UnifOncust 22784  unifTopcutop 22815  UnifStcuss 22838  UnifSpcusp 22839   Cnucucn 22860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-map 8386  df-top 21478  df-topon 21495  df-topsp 21517  df-cn 21811  df-ust 22785  df-utop 22816  df-usp 22842  df-ucn 22861 This theorem is referenced by: (None)
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