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Theorem ucncn 22821
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 2818 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
4 eqid 2818 . . . . . . . 8 (UnifSt‘𝑅) = (UnifSt‘𝑅)
5 ucncn.j . . . . . . . 8 𝐽 = (TopOpen‘𝑅)
63, 4, 5isusp 22797 . . . . . . 7 (𝑅 ∈ UnifSp ↔ ((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑅))))
76simplbi 498 . . . . . 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 2818 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
11 eqid 2818 . . . . . . . 8 (UnifSt‘𝑆) = (UnifSt‘𝑆)
12 ucncn.k . . . . . . . 8 𝐾 = (TopOpen‘𝑆)
1310, 11, 12isusp 22797 . . . . . . 7 (𝑆 ∈ UnifSp ↔ ((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) ∧ 𝐾 = (unifTop‘(UnifSt‘𝑆))))
1413simplbi 498 . . . . . 6 (𝑆 ∈ UnifSp → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
159, 14syl 17 . . . . 5 (𝜑 → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
16 isucn 22814 . . . . 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 584 . . . 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 233 . . 3 (𝜑 → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))))
1918simpld 495 . 2 (𝜑𝐹:(Base‘𝑅)⟶(Base‘𝑆))
20 cnvimass 5942 . . . . 5 (𝐹𝑎) ⊆ dom 𝐹
2119fdmd 6516 . . . . . 6 (𝜑 → dom 𝐹 = (Base‘𝑅))
2221adantr 481 . . . . 5 ((𝜑𝑎𝐾) → dom 𝐹 = (Base‘𝑅))
2320, 22sseqtrid 4016 . . . 4 ((𝜑𝑎𝐾) → (𝐹𝑎) ⊆ (Base‘𝑅))
24 simplll 771 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝜑)
25 simpr 485 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑠 ∈ (UnifSt‘𝑆))
2623ad2antrr 722 . . . . . . . . . 10 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → (𝐹𝑎) ⊆ (Base‘𝑅))
27 simplr 765 . . . . . . . . . 10 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑥 ∈ (𝐹𝑎))
2826, 27sseldd 3965 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
2918simprd 496 . . . . . . . . . . . 12 (𝜑 → ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3029r19.21bi 3205 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (UnifSt‘𝑆)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
31 r19.12 3321 . . . . . . . . . . 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 3205 . . . . . . . . 9 (((𝜑𝑠 ∈ (UnifSt‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3424, 25, 28, 33syl21anc 833 . . . . . . . 8 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3534adantr 481 . . . . . . 7 (((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3624ad3antrrr 726 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝜑)
378ad5antr 730 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)))
38 simpr 485 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → 𝑟 ∈ (UnifSt‘𝑅))
39 ustrel 22747 . . . . . . . . . . . 12 (((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑟)
4037, 38, 39syl2anc 584 . . . . . . . . . . 11 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑟)
4140adantr 481 . . . . . . . . . 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 765 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝑟 ∈ (UnifSt‘𝑅))
4428ad3antrrr 726 . . . . . . . . . . 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 22764 . . . . . . . . . . 11 (((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝑟 ∈ (UnifSt‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
4642, 43, 44, 45syl3anc 1363 . . . . . . . . . 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 485 . . . . . . . . . . 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 765 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → 𝑧 ∈ (Base‘𝑅))
49 simpllr 772 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎)
5015ad5antr 730 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
51 simpllr 772 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → 𝑠 ∈ (UnifSt‘𝑆))
52 ustrel 22747 . . . . . . . . . . . . . . . . . . . 20 (((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → Rel 𝑠)
5350, 51, 52syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑠)
54 elrelimasn 5946 . . . . . . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ (𝑠 “ {(𝐹𝑥)}))
5749, 56sseldd 3965 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ 𝑎)
5857adantlr 711 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ 𝑎)
59 ffn 6507 . . . . . . . . . . . . . . . . 17 (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → 𝐹 Fn (Base‘𝑅))
60 elpreima 6820 . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . 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 709 . . . . . . . . . . . . . 14 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → 𝑧 ∈ (𝐹𝑎))
6463ex 413 . . . . . . . . . . . . 13 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) → ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
6564ralrimiva 3179 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
6665adantr 481 . . . . . . . . . . 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 3167 . . . . . . . . . . . 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 pm3.33 761 . . . . . . . . . . . . 13 (((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → (𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
6968ralimi 3157 . . . . . . . . . . . 12 (∀𝑧 ∈ (Base‘𝑅)((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
7067, 69sylbir 236 . . . . . . . . . . 11 ((∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
7147, 66, 70syl2anc 584 . . . . . . . . . 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 1218 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → Rel 𝑟)
73 simpr 485 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (𝑟 “ {𝑥}))
74 elrelimasn 5946 . . . . . . . . . . . . . . 15 (Rel 𝑟 → (𝑦 ∈ (𝑟 “ {𝑥}) ↔ 𝑥𝑟𝑦))
7574biimpa 477 . . . . . . . . . . . . . 14 ((Rel 𝑟𝑦 ∈ (𝑟 “ {𝑥})) → 𝑥𝑟𝑦)
7672, 73, 75syl2anc 584 . . . . . . . . . . . . 13 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑥𝑟𝑦)
77 breq2 5061 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑥𝑟𝑧𝑥𝑟𝑦))
78 eleq1w 2892 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑧 ∈ (𝐹𝑎) ↔ 𝑦 ∈ (𝐹𝑎)))
7977, 78imbi12d 346 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)) ↔ (𝑥𝑟𝑦𝑦 ∈ (𝐹𝑎))))
80 simpl3 1185 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
81 simpl2r 1219 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
8281, 73sseldd 3965 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (Base‘𝑅))
8379, 80, 82rspcdva 3622 . . . . . . . . . . . . 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 413 . . . . . . . . . . 11 ((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) → (𝑦 ∈ (𝑟 “ {𝑥}) → 𝑦 ∈ (𝐹𝑎)))
8685ssrdv 3970 . . . . . . . . . 10 ((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
8736, 41, 46, 71, 86syl121anc 1367 . . . . . . . . 9 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
8887ex 413 . . . . . . . 8 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎)))
8988reximdva 3271 . . . . . . 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 4567 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → {𝑦} = {(𝐹𝑥)})
9291imaeq2d 5922 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → (𝑠 “ {𝑦}) = (𝑠 “ {(𝐹𝑥)}))
9392sseq1d 3995 . . . . . . . 8 (𝑦 = (𝐹𝑥) → ((𝑠 “ {𝑦}) ⊆ 𝑎 ↔ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
9493rexbidv 3294 . . . . . . 7 (𝑦 = (𝐹𝑥) → (∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎 ↔ ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
95 simpr 485 . . . . . . . . . . 11 ((𝜑𝑎𝐾) → 𝑎𝐾)
9613simprbi 497 . . . . . . . . . . . . 13 (𝑆 ∈ UnifSp → 𝐾 = (unifTop‘(UnifSt‘𝑆)))
979, 96syl 17 . . . . . . . . . . . 12 (𝜑𝐾 = (unifTop‘(UnifSt‘𝑆)))
9897adantr 481 . . . . . . . . . . 11 ((𝜑𝑎𝐾) → 𝐾 = (unifTop‘(UnifSt‘𝑆)))
9995, 98eleqtrd 2912 . . . . . . . . . 10 ((𝜑𝑎𝐾) → 𝑎 ∈ (unifTop‘(UnifSt‘𝑆)))
100 elutop 22769 . . . . . . . . . . . 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 481 . . . . . . . . . 10 ((𝜑𝑎𝐾) → (𝑎 ∈ (unifTop‘(UnifSt‘𝑆)) ↔ (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)))
10399, 102mpbid 233 . . . . . . . . 9 ((𝜑𝑎𝐾) → (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎))
104103simprd 496 . . . . . . . 8 ((𝜑𝑎𝐾) → ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)
105104adantr 481 . . . . . . 7 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)
106 elpreima 6820 . . . . . . . . . . 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 481 . . . . . . . . 9 ((𝜑𝑎𝐾) → (𝑥 ∈ (𝐹𝑎) ↔ (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎)))
109108biimpa 477 . . . . . . . 8 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎))
110109simprd 496 . . . . . . 7 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → (𝐹𝑥) ∈ 𝑎)
11194, 105, 110rspcdva 3622 . . . . . 6 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎)
11290, 111r19.29a 3286 . . . . 5 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
113112ralrimiva 3179 . . . 4 ((𝜑𝑎𝐾) → ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
1146simprbi 497 . . . . . . . 8 (𝑅 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑅)))
1152, 114syl 17 . . . . . . 7 (𝜑𝐽 = (unifTop‘(UnifSt‘𝑅)))
116115adantr 481 . . . . . 6 ((𝜑𝑎𝐾) → 𝐽 = (unifTop‘(UnifSt‘𝑅)))
117116eleq2d 2895 . . . . 5 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ 𝐽 ↔ (𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅))))
118 elutop 22769 . . . . . . 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 481 . . . . 5 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅)) ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
121117, 120bitrd 280 . . . 4 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ 𝐽 ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
12223, 113, 121mpbir2and 709 . . 3 ((𝜑𝑎𝐾) → (𝐹𝑎) ∈ 𝐽)
123122ralrimiva 3179 . 2 (𝜑 → ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)
124 ucncn.3 . . . 4 (𝜑𝑅 ∈ TopSp)
1253, 5istps 21470 . . . 4 (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑅)))
126124, 125sylib 219 . . 3 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑅)))
127 ucncn.4 . . . 4 (𝜑𝑆 ∈ TopSp)
12810, 12istps 21470 . . . 4 (𝑆 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝑆)))
129127, 128sylib 219 . . 3 (𝜑𝐾 ∈ (TopOn‘(Base‘𝑆)))
130 iscn 21771 . . 3 ((𝐽 ∈ (TopOn‘(Base‘𝑅)) ∧ 𝐾 ∈ (TopOn‘(Base‘𝑆))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)))
131126, 129, 130syl2anc 584 . 2 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)))
13219, 123, 131mpbir2and 709 1 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933  {csn 4557   class class class wbr 5057  ccnv 5547  dom cdm 5548  cima 5551  Rel wrel 5553   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  Basecbs 16471  TopOpenctopn 16683  TopOnctopon 21446  TopSpctps 21468   Cn ccn 21760  UnifOncust 22735  unifTopcutop 22766  UnifStcuss 22789  UnifSpcusp 22790   Cnucucn 22811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397  df-top 21430  df-topon 21447  df-topsp 21469  df-cn 21763  df-ust 22736  df-utop 22767  df-usp 22793  df-ucn 22812
This theorem is referenced by: (None)
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