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Theorem ucncn 21994
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 2626 . . . . . . . 8 (Base‘𝑅) = (Base‘𝑅)
4 eqid 2626 . . . . . . . 8 (UnifSt‘𝑅) = (UnifSt‘𝑅)
5 ucncn.j . . . . . . . 8 𝐽 = (TopOpen‘𝑅)
63, 4, 5isusp 21970 . . . . . . 7 (𝑅 ∈ UnifSp ↔ ((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝐽 = (unifTop‘(UnifSt‘𝑅))))
76simplbi 476 . . . . . 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 2626 . . . . . . . 8 (Base‘𝑆) = (Base‘𝑆)
11 eqid 2626 . . . . . . . 8 (UnifSt‘𝑆) = (UnifSt‘𝑆)
12 ucncn.k . . . . . . . 8 𝐾 = (TopOpen‘𝑆)
1310, 11, 12isusp 21970 . . . . . . 7 (𝑆 ∈ UnifSp ↔ ((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) ∧ 𝐾 = (unifTop‘(UnifSt‘𝑆))))
1413simplbi 476 . . . . . 6 (𝑆 ∈ UnifSp → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
159, 14syl 17 . . . . 5 (𝜑 → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
16 isucn 21987 . . . . 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 692 . . . 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 222 . . 3 (𝜑 → (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))))
1918simpld 475 . 2 (𝜑𝐹:(Base‘𝑅)⟶(Base‘𝑆))
20 cnvimass 5448 . . . . 5 (𝐹𝑎) ⊆ dom 𝐹
21 fdm 6010 . . . . . . 7 (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → dom 𝐹 = (Base‘𝑅))
2219, 21syl 17 . . . . . 6 (𝜑 → dom 𝐹 = (Base‘𝑅))
2322adantr 481 . . . . 5 ((𝜑𝑎𝐾) → dom 𝐹 = (Base‘𝑅))
2420, 23syl5sseq 3637 . . . 4 ((𝜑𝑎𝐾) → (𝐹𝑎) ⊆ (Base‘𝑅))
25 simplll 797 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝜑)
26 simpr 477 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑠 ∈ (UnifSt‘𝑆))
2724ad2antrr 761 . . . . . . . . . 10 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → (𝐹𝑎) ⊆ (Base‘𝑅))
28 simplr 791 . . . . . . . . . 10 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑥 ∈ (𝐹𝑎))
2927, 28sseldd 3589 . . . . . . . . 9 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → 𝑥 ∈ (Base‘𝑅))
3018simprd 479 . . . . . . . . . . . 12 (𝜑 → ∀𝑠 ∈ (UnifSt‘𝑆)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3130r19.21bi 2932 . . . . . . . . . . 11 ((𝜑𝑠 ∈ (UnifSt‘𝑆)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑥 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
32 r19.12 3061 . . . . . . . . . . 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)))
3331, 32syl 17 . . . . . . . . . 10 ((𝜑𝑠 ∈ (UnifSt‘𝑆)) → ∀𝑥 ∈ (Base‘𝑅)∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3433r19.21bi 2932 . . . . . . . . 9 (((𝜑𝑠 ∈ (UnifSt‘𝑆)) ∧ 𝑥 ∈ (Base‘𝑅)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3525, 26, 29, 34syl21anc 1322 . . . . . . . 8 ((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3635adantr 481 . . . . . . 7 (((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) → ∃𝑟 ∈ (UnifSt‘𝑅)∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)))
3725ad3antrrr 765 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝜑)
388ad5antr 769 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)))
39 simpr 477 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → 𝑟 ∈ (UnifSt‘𝑅))
40 ustrel 21920 . . . . . . . . . . . 12 (((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑟)
4138, 39, 40syl2anc 692 . . . . . . . . . . 11 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑟)
4241adantr 481 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → Rel 𝑟)
4337, 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)))
44 simplr 791 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝑟 ∈ (UnifSt‘𝑅))
4529ad3antrrr 765 . . . . . . . . . . 11 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → 𝑥 ∈ (Base‘𝑅))
46 ustimasn 21937 . . . . . . . . . . 11 (((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) ∧ 𝑟 ∈ (UnifSt‘𝑅) ∧ 𝑥 ∈ (Base‘𝑅)) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
4743, 44, 45, 46syl3anc 1323 . . . . . . . . . 10 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
48 simpr 477 . . . . . . . . . . 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)))
49 simplr 791 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → 𝑧 ∈ (Base‘𝑅))
50 simpllr 798 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎)
5115ad5antr 769 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)))
52 simpllr 798 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → 𝑠 ∈ (UnifSt‘𝑆))
53 ustrel 21920 . . . . . . . . . . . . . . . . . . . 20 (((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) → Rel 𝑠)
5451, 52, 53syl2anc 692 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → Rel 𝑠)
55 elrelimasn 5452 . . . . . . . . . . . . . . . . . . 19 (Rel 𝑠 → ((𝐹𝑧) ∈ (𝑠 “ {(𝐹𝑥)}) ↔ (𝐹𝑥)𝑠(𝐹𝑧)))
5654, 55syl 17 . . . . . . . . . . . . . . . . . 18 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → ((𝐹𝑧) ∈ (𝑠 “ {(𝐹𝑥)}) ↔ (𝐹𝑥)𝑠(𝐹𝑧)))
5756biimpar 502 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ (𝑠 “ {(𝐹𝑥)}))
5850, 57sseldd 3589 . . . . . . . . . . . . . . . 16 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ 𝑎)
5958adantlr 750 . . . . . . . . . . . . . . 15 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → (𝐹𝑧) ∈ 𝑎)
60 ffn 6004 . . . . . . . . . . . . . . . . 17 (𝐹:(Base‘𝑅)⟶(Base‘𝑆) → 𝐹 Fn (Base‘𝑅))
61 elpreima 6294 . . . . . . . . . . . . . . . . 17 (𝐹 Fn (Base‘𝑅) → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧 ∈ (Base‘𝑅) ∧ (𝐹𝑧) ∈ 𝑎)))
6219, 60, 613syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑧 ∈ (𝐹𝑎) ↔ (𝑧 ∈ (Base‘𝑅) ∧ (𝐹𝑧) ∈ 𝑎)))
6362ad7antr 773 . . . . . . . . . . . . . . 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)))
6449, 59, 63mpbir2and 956 . . . . . . . . . . . . . 14 ((((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝐹𝑥)𝑠(𝐹𝑧)) → 𝑧 ∈ (𝐹𝑎))
6564ex 450 . . . . . . . . . . . . 13 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ 𝑧 ∈ (Base‘𝑅)) → ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
6665ralrimiva 2965 . . . . . . . . . . . 12 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎)))
6766adantr 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)))
68 r19.26 3062 . . . . . . . . . . . 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))))
69 pm3.33 608 . . . . . . . . . . . . 13 (((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → (𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
7069ralimi 2952 . . . . . . . . . . . 12 (∀𝑧 ∈ (Base‘𝑅)((𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
7168, 70sylbir 225 . . . . . . . . . . 11 ((∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) ∧ ∀𝑧 ∈ (Base‘𝑅)((𝐹𝑥)𝑠(𝐹𝑧) → 𝑧 ∈ (𝐹𝑎))) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
7248, 67, 71syl2anc 692 . . . . . . . . . 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)))
73 simpl2l 1112 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → Rel 𝑟)
74 simpr 477 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (𝑟 “ {𝑥}))
75 elrelimasn 5452 . . . . . . . . . . . . . . 15 (Rel 𝑟 → (𝑦 ∈ (𝑟 “ {𝑥}) ↔ 𝑥𝑟𝑦))
7675biimpa 501 . . . . . . . . . . . . . 14 ((Rel 𝑟𝑦 ∈ (𝑟 “ {𝑥})) → 𝑥𝑟𝑦)
7773, 74, 76syl2anc 692 . . . . . . . . . . . . 13 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑥𝑟𝑦)
78 simpl2r 1113 . . . . . . . . . . . . . . 15 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → (𝑟 “ {𝑥}) ⊆ (Base‘𝑅))
7978, 74sseldd 3589 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (Base‘𝑅))
80 simpl3 1064 . . . . . . . . . . . . . 14 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)))
81 breq2 4622 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (𝑥𝑟𝑧𝑥𝑟𝑦))
82 eleq1 2692 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑦 → (𝑧 ∈ (𝐹𝑎) ↔ 𝑦 ∈ (𝐹𝑎)))
8381, 82imbi12d 334 . . . . . . . . . . . . . . 15 (𝑧 = 𝑦 → ((𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)) ↔ (𝑥𝑟𝑦𝑦 ∈ (𝐹𝑎))))
8483rspcv 3296 . . . . . . . . . . . . . 14 (𝑦 ∈ (Base‘𝑅) → (∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎)) → (𝑥𝑟𝑦𝑦 ∈ (𝐹𝑎))))
8579, 80, 84sylc 65 . . . . . . . . . . . . 13 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → (𝑥𝑟𝑦𝑦 ∈ (𝐹𝑎)))
8677, 85mpd 15 . . . . . . . . . . . 12 (((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) ∧ 𝑦 ∈ (𝑟 “ {𝑥})) → 𝑦 ∈ (𝐹𝑎))
8786ex 450 . . . . . . . . . . 11 ((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) → (𝑦 ∈ (𝑟 “ {𝑥}) → 𝑦 ∈ (𝐹𝑎)))
8887ssrdv 3594 . . . . . . . . . 10 ((𝜑 ∧ (Rel 𝑟 ∧ (𝑟 “ {𝑥}) ⊆ (Base‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧𝑧 ∈ (𝐹𝑎))) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
8937, 42, 47, 72, 88syl121anc 1328 . . . . . . . . 9 (((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) ∧ ∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧))) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
9089ex 450 . . . . . . . 8 ((((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) ∧ 𝑟 ∈ (UnifSt‘𝑅)) → (∀𝑧 ∈ (Base‘𝑅)(𝑥𝑟𝑧 → (𝐹𝑥)𝑠(𝐹𝑧)) → (𝑟 “ {𝑥}) ⊆ (𝐹𝑎)))
9190reximdva 3016 . . . . . . 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)))
9236, 91mpd 15 . . . . . 6 (((((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) ∧ 𝑠 ∈ (UnifSt‘𝑆)) ∧ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎) → ∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
93 elpreima 6294 . . . . . . . . . . 11 (𝐹 Fn (Base‘𝑅) → (𝑥 ∈ (𝐹𝑎) ↔ (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎)))
9419, 60, 933syl 18 . . . . . . . . . 10 (𝜑 → (𝑥 ∈ (𝐹𝑎) ↔ (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎)))
9594adantr 481 . . . . . . . . 9 ((𝜑𝑎𝐾) → (𝑥 ∈ (𝐹𝑎) ↔ (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎)))
9695biimpa 501 . . . . . . . 8 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → (𝑥 ∈ (Base‘𝑅) ∧ (𝐹𝑥) ∈ 𝑎))
9796simprd 479 . . . . . . 7 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → (𝐹𝑥) ∈ 𝑎)
98 simpr 477 . . . . . . . . . . 11 ((𝜑𝑎𝐾) → 𝑎𝐾)
9913simprbi 480 . . . . . . . . . . . . 13 (𝑆 ∈ UnifSp → 𝐾 = (unifTop‘(UnifSt‘𝑆)))
1009, 99syl 17 . . . . . . . . . . . 12 (𝜑𝐾 = (unifTop‘(UnifSt‘𝑆)))
101100adantr 481 . . . . . . . . . . 11 ((𝜑𝑎𝐾) → 𝐾 = (unifTop‘(UnifSt‘𝑆)))
10298, 101eleqtrd 2706 . . . . . . . . . 10 ((𝜑𝑎𝐾) → 𝑎 ∈ (unifTop‘(UnifSt‘𝑆)))
103 elutop 21942 . . . . . . . . . . . 12 ((UnifSt‘𝑆) ∈ (UnifOn‘(Base‘𝑆)) → (𝑎 ∈ (unifTop‘(UnifSt‘𝑆)) ↔ (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)))
10415, 103syl 17 . . . . . . . . . . 11 (𝜑 → (𝑎 ∈ (unifTop‘(UnifSt‘𝑆)) ↔ (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)))
105104adantr 481 . . . . . . . . . 10 ((𝜑𝑎𝐾) → (𝑎 ∈ (unifTop‘(UnifSt‘𝑆)) ↔ (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)))
106102, 105mpbid 222 . . . . . . . . 9 ((𝜑𝑎𝐾) → (𝑎 ⊆ (Base‘𝑆) ∧ ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎))
107106simprd 479 . . . . . . . 8 ((𝜑𝑎𝐾) → ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)
108107adantr 481 . . . . . . 7 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎)
109 sneq 4163 . . . . . . . . . . 11 (𝑦 = (𝐹𝑥) → {𝑦} = {(𝐹𝑥)})
110109imaeq2d 5429 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → (𝑠 “ {𝑦}) = (𝑠 “ {(𝐹𝑥)}))
111110sseq1d 3616 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝑠 “ {𝑦}) ⊆ 𝑎 ↔ (𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
112111rexbidv 3050 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎 ↔ ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
113112rspcv 3296 . . . . . . 7 ((𝐹𝑥) ∈ 𝑎 → (∀𝑦𝑎𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {𝑦}) ⊆ 𝑎 → ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎))
11497, 108, 113sylc 65 . . . . . 6 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∃𝑠 ∈ (UnifSt‘𝑆)(𝑠 “ {(𝐹𝑥)}) ⊆ 𝑎)
11592, 114r19.29a 3076 . . . . 5 (((𝜑𝑎𝐾) ∧ 𝑥 ∈ (𝐹𝑎)) → ∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
116115ralrimiva 2965 . . . 4 ((𝜑𝑎𝐾) → ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))
1176simprbi 480 . . . . . . . 8 (𝑅 ∈ UnifSp → 𝐽 = (unifTop‘(UnifSt‘𝑅)))
1182, 117syl 17 . . . . . . 7 (𝜑𝐽 = (unifTop‘(UnifSt‘𝑅)))
119118adantr 481 . . . . . 6 ((𝜑𝑎𝐾) → 𝐽 = (unifTop‘(UnifSt‘𝑅)))
120119eleq2d 2689 . . . . 5 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ 𝐽 ↔ (𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅))))
121 elutop 21942 . . . . . . 7 ((UnifSt‘𝑅) ∈ (UnifOn‘(Base‘𝑅)) → ((𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅)) ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
1228, 121syl 17 . . . . . 6 (𝜑 → ((𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅)) ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
123122adantr 481 . . . . 5 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ (unifTop‘(UnifSt‘𝑅)) ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
124120, 123bitrd 268 . . . 4 ((𝜑𝑎𝐾) → ((𝐹𝑎) ∈ 𝐽 ↔ ((𝐹𝑎) ⊆ (Base‘𝑅) ∧ ∀𝑥 ∈ (𝐹𝑎)∃𝑟 ∈ (UnifSt‘𝑅)(𝑟 “ {𝑥}) ⊆ (𝐹𝑎))))
12524, 116, 124mpbir2and 956 . . 3 ((𝜑𝑎𝐾) → (𝐹𝑎) ∈ 𝐽)
126125ralrimiva 2965 . 2 (𝜑 → ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)
127 ucncn.3 . . . 4 (𝜑𝑅 ∈ TopSp)
1283, 5istps 20646 . . . 4 (𝑅 ∈ TopSp ↔ 𝐽 ∈ (TopOn‘(Base‘𝑅)))
129127, 128sylib 208 . . 3 (𝜑𝐽 ∈ (TopOn‘(Base‘𝑅)))
130 ucncn.4 . . . 4 (𝜑𝑆 ∈ TopSp)
13110, 12istps 20646 . . . 4 (𝑆 ∈ TopSp ↔ 𝐾 ∈ (TopOn‘(Base‘𝑆)))
132130, 131sylib 208 . . 3 (𝜑𝐾 ∈ (TopOn‘(Base‘𝑆)))
133 iscn 20944 . . 3 ((𝐽 ∈ (TopOn‘(Base‘𝑅)) ∧ 𝐾 ∈ (TopOn‘(Base‘𝑆))) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)))
134129, 132, 133syl2anc 692 . 2 (𝜑 → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:(Base‘𝑅)⟶(Base‘𝑆) ∧ ∀𝑎𝐾 (𝐹𝑎) ∈ 𝐽)))
13519, 126, 134mpbir2and 956 1 (𝜑𝐹 ∈ (𝐽 Cn 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1992  wral 2912  wrex 2913  wss 3560  {csn 4153   class class class wbr 4618  ccnv 5078  dom cdm 5079  cima 5082  Rel wrel 5084   Fn wfn 5845  wf 5846  cfv 5850  (class class class)co 6605  Basecbs 15776  TopOpenctopn 15998  TopOnctopon 20613  TopSpctps 20614   Cn ccn 20933  UnifOncust 21908  unifTopcutop 21939  UnifStcuss 21962  UnifSpcusp 21963   Cnucucn 21984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-fv 5858  df-ov 6608  df-oprab 6609  df-mpt2 6610  df-map 7805  df-top 20616  df-topon 20618  df-topsp 20619  df-cn 20936  df-ust 21909  df-utop 21940  df-usp 21966  df-ucn 21985
This theorem is referenced by: (None)
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