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Theorem ucncn 24294
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 2737 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
4 eqid 2737 . . . . . . . 8 (UnifSt‘𝑅) = (UnifSt‘𝑅)
5 ucncn.j . . . . . . . 8 𝐽 = (TopOpen‘𝑅)
63, 4, 5isusp 24270 . . . . . . 7 (𝑅 ∈ UnifSp ↔ ((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑅))))
76simplbi 497 . . . . . 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 2737 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
11 eqid 2737 . . . . . . . 8 (UnifSt‘𝑆) = (UnifSt‘𝑆)
12 ucncn.k . . . . . . . 8 𝐾 = (TopOpen‘𝑆)
1310, 11, 12isusp 24270 . . . . . . 7 (𝑆 ∈ UnifSp ↔ ((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) ∧ 𝐾 = (unifTop‘(UnifSt‘𝑆))))
1413simplbi 497 . . . . . 6 (𝑆 ∈ UnifSp → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
159, 14syl 17 . . . . 5 (𝜑 → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
16 isucn 24287 . . . . 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 232 . . 3 (𝜑 → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))))
1918simpld 494 . 2 (𝜑𝐹:(Base‘𝑅)⟶(Base‘𝑆))
20 cnvimass 6100 . . . . 5 (𝐹𝑎) ⊆ dom 𝐹
2119fdmd 6746 . . . . . 6 (𝜑 → dom 𝐹 = (Base‘𝑅))
2221adantr 480 . . . . 5 ((𝜑𝑎𝐾) → dom 𝐹 = (Base‘𝑅))
2320, 22sseqtrid 4026 . . . 4 ((𝜑𝑎𝐾) → (𝐹𝑎) ⊆ (Base‘𝑅))
24 simplll 775 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝜑)
25 simpr 484 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑠 ∈ (UnifSt‘𝑆))
2623ad2antrr 726 . . . . . . . . . 10 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → (𝐹𝑎) ⊆ (Base‘𝑅))
27 simplr 769 . . . . . . . . . 10 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑥 ∈ (𝐹𝑎))
2826, 27sseldd 3984 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
2918simprd 495 . . . . . . . . . . . 12 (𝜑 → ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3029r19.21bi 3251 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (UnifSt‘𝑆)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
31 r19.12 3314 . . . . . . . . . . 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 3251 . . . . . . . . 9 (((𝜑𝑠 ∈ (UnifSt‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3424, 25, 28, 33syl21anc 838 . . . . . . . 8 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3534adantr 480 . . . . . . 7 (((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3624ad3antrrr 730 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝜑)
378ad5antr 734 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)))
38 simpr 484 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → 𝑟 ∈ (UnifSt‘𝑅))
39 ustrel 24220 . . . . . . . . . . . 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 480 . . . . . . . . . 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 769 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝑟 ∈ (UnifSt‘𝑅))
4428ad3antrrr 730 . . . . . . . . . . 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 24237 . . . . . . . . . . 11 (((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝑟 ∈ (UnifSt‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
4642, 43, 44, 45syl3anc 1373 . . . . . . . . . 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 484 . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → 𝑧 ∈ (Base‘𝑅))
49 simpllr 776 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎)
5015ad5antr 734 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
51 simpllr 776 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → 𝑠 ∈ (UnifSt‘𝑆))
52 ustrel 24220 . . . . . . . . . . . . . . . . . . . 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 6104 . . . . . . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ (𝑠 “ {(𝐹𝑥)}))
5749, 56sseldd 3984 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ 𝑎)
5857adantlr 715 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ 𝑎)
59 ffn 6736 . . . . . . . . . . . . . . . . 17 (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → 𝐹 Fn (Base‘𝑅))
60 elpreima 7078 . . . . . . . . . . . . . . . . 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 738 . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . 14 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → 𝑧 ∈ (𝐹𝑎))
6463ex 412 . . . . . . . . . . . . 13 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) → ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
6564ralrimiva 3146 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
6665adantr 480 . . . . . . . . . . 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 3111 . . . . . . . . . . . 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 pm3.33 765 . . . . . . . . . . . . 13 (((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → (𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
6968ralimi 3083 . . . . . . . . . . . 12 (∀𝑧 ∈ (Base‘𝑅)((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
7067, 69sylbir 235 . . . . . . . . . . 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 1227 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → Rel 𝑟)
73 simpr 484 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (𝑟 “ {𝑥}))
74 elrelimasn 6104 . . . . . . . . . . . . . . 15 (Rel 𝑟 → (𝑦 ∈ (𝑟 “ {𝑥}) ↔ 𝑥𝑟𝑦))
7574biimpa 476 . . . . . . . . . . . . . 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 5147 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑥𝑟𝑧𝑥𝑟𝑦))
78 eleq1w 2824 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → (𝑧 ∈ (𝐹𝑎) ↔ 𝑦 ∈ (𝐹𝑎)))
7977, 78imbi12d 344 . . . . . . . . . . . . . 14 (𝑧 = 𝑦 → ((𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)) ↔ (𝑥𝑟𝑦𝑦 ∈ (𝐹𝑎))))
80 simpl3 1194 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
81 simpl2r 1228 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
8281, 73sseldd 3984 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (Base‘𝑅))
8379, 80, 82rspcdva 3623 . . . . . . . . . . . . 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 412 . . . . . . . . . . 11 ((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) → (𝑦 ∈ (𝑟 “ {𝑥}) → 𝑦 ∈ (𝐹𝑎)))
8685ssrdv 3989 . . . . . . . . . 10 ((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
8736, 41, 46, 71, 86syl121anc 1377 . . . . . . . . 9 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
8887ex 412 . . . . . . . 8 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎)))
8988reximdva 3168 . . . . . . 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 4636 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → {𝑦} = {(𝐹𝑥)})
9291imaeq2d 6078 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → (𝑠 “ {𝑦}) = (𝑠 “ {(𝐹𝑥)}))
9392sseq1d 4015 . . . . . . . 8 (𝑦 = (𝐹𝑥) → ((𝑠 “ {𝑦}) ⊆ 𝑎 ↔ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
9493rexbidv 3179 . . . . . . 7 (𝑦 = (𝐹𝑥) → (∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎 ↔ ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
95 simpr 484 . . . . . . . . . . 11 ((𝜑𝑎𝐾) → 𝑎𝐾)
9613simprbi 496 . . . . . . . . . . . . 13 (𝑆 ∈ UnifSp → 𝐾 = (unifTop‘(UnifSt‘𝑆)))
979, 96syl 17 . . . . . . . . . . . 12 (𝜑𝐾 = (unifTop‘(UnifSt‘𝑆)))
9897adantr 480 . . . . . . . . . . 11 ((𝜑𝑎𝐾) → 𝐾 = (unifTop‘(UnifSt‘𝑆)))
9995, 98eleqtrd 2843 . . . . . . . . . 10 ((𝜑𝑎𝐾) → 𝑎 ∈ (unifTop‘(UnifSt‘𝑆)))
100 elutop 24242 . . . . . . . . . . . 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 480 . . . . . . . . . 10 ((𝜑𝑎𝐾) → (𝑎 ∈ (unifTop‘(UnifSt‘𝑆)) ↔ (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)))
10399, 102mpbid 232 . . . . . . . . 9 ((𝜑𝑎𝐾) → (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎))
104103simprd 495 . . . . . . . 8 ((𝜑𝑎𝐾) → ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)
105104adantr 480 . . . . . . 7 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)
106 elpreima 7078 . . . . . . . . . . 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 480 . . . . . . . . 9 ((𝜑𝑎𝐾) → (𝑥 ∈ (𝐹𝑎) ↔ (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎)))
109108biimpa 476 . . . . . . . 8 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎))
110109simprd 495 . . . . . . 7 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → (𝐹𝑥) ∈ 𝑎)
11194, 105, 110rspcdva 3623 . . . . . 6 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎)
11290, 111r19.29a 3162 . . . . 5 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
113112ralrimiva 3146 . . . 4 ((𝜑𝑎𝐾) → ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
1146simprbi 496 . . . . . . . 8 (𝑅 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑅)))
1152, 114syl 17 . . . . . . 7 (𝜑𝐽 = (unifTop‘(UnifSt‘𝑅)))
116115adantr 480 . . . . . 6 ((𝜑𝑎𝐾) → 𝐽 = (unifTop‘(UnifSt‘𝑅)))
117116eleq2d 2827 . . . . 5 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ 𝐽 ↔ (𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅))))
118 elutop 24242 . . . . . . 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 480 . . . . 5 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅)) ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
121117, 120bitrd 279 . . . 4 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ 𝐽 ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
12223, 113, 121mpbir2and 713 . . 3 ((𝜑𝑎𝐾) → (𝐹𝑎) ∈ 𝐽)
123122ralrimiva 3146 . 2 (𝜑 → ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)
124 ucncn.3 . . . 4 (𝜑𝑅 ∈ TopSp)
1253, 5istps 22940 . . . 4 (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑅)))
126124, 125sylib 218 . . 3 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑅)))
127 ucncn.4 . . . 4 (𝜑𝑆 ∈ TopSp)
12810, 12istps 22940 . . . 4 (𝑆 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝑆)))
129127, 128sylib 218 . . 3 (𝜑𝐾 ∈ (TopOn‘(Base‘𝑆)))
130 iscn 23243 . . 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 713 1 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951  {csn 4626   class class class wbr 5143  ccnv 5684  dom cdm 5685  cima 5688  Rel wrel 5690   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  TopOpenctopn 17466  TopOnctopon 22916  TopSpctps 22938   Cn ccn 23232  UnifOncust 24208  unifTopcutop 24239  UnifStcuss 24262  UnifSpcusp 24263   Cnucucn 24284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-top 22900  df-topon 22917  df-topsp 22939  df-cn 23235  df-ust 24209  df-utop 24240  df-usp 24266  df-ucn 24285
This theorem is referenced by: (None)
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