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Theorem isucn2 22892
 Description: The predicate "𝐹 is a uniformly continuous function from uniform space 𝑈 to uniform space 𝑉", expressed with filter bases for the entourages. (Contributed by Thierry Arnoux, 26-Jan-2018.)
Hypotheses
Ref Expression
isucn2.u 𝑈 = ((𝑋 × 𝑋)filGen𝑅)
isucn2.v 𝑉 = ((𝑌 × 𝑌)filGen𝑆)
isucn2.1 (𝜑𝑈 ∈ (UnifOn‘𝑋))
isucn2.2 (𝜑𝑉 ∈ (UnifOn‘𝑌))
isucn2.3 (𝜑𝑅 ∈ (fBas‘(𝑋 × 𝑋)))
isucn2.4 (𝜑𝑆 ∈ (fBas‘(𝑌 × 𝑌)))
Assertion
Ref Expression
isucn2 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
Distinct variable groups:   𝑠,𝑟,𝑥,𝑦,𝐹   𝑅,𝑟,𝑥,𝑦   𝑆,𝑠,𝑥,𝑦   𝑈,𝑟,𝑠,𝑥,𝑦   𝑉,𝑠,𝑥   𝑋,𝑟,𝑠,𝑥,𝑦   𝑌,𝑠,𝑥,𝑦   𝜑,𝑟,𝑠,𝑥,𝑦
Allowed substitution hints:   𝑅(𝑠)   𝑆(𝑟)   𝑉(𝑦,𝑟)   𝑌(𝑟)

Proof of Theorem isucn2
Dummy variables 𝑢 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isucn2.1 . . 3 (𝜑𝑈 ∈ (UnifOn‘𝑋))
2 isucn2.2 . . 3 (𝜑𝑉 ∈ (UnifOn‘𝑌))
3 isucn 22891 . . 3 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ (UnifOn‘𝑌)) → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))))
41, 2, 3syl2anc 587 . 2 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))))
5 breq 5032 . . . . . . . . . 10 (𝑣 = 𝑠 → ((𝐹𝑥)𝑣(𝐹𝑦) ↔ (𝐹𝑥)𝑠(𝐹𝑦)))
65imbi2d 344 . . . . . . . . 9 (𝑣 = 𝑠 → ((𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
76ralbidv 3162 . . . . . . . 8 (𝑣 = 𝑠 → (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
87rexralbidv 3260 . . . . . . 7 (𝑣 = 𝑠 → (∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
9 simplr 768 . . . . . . 7 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) ∧ 𝑠𝑆) → ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))
10 isucn2.4 . . . . . . . . . . . 12 (𝜑𝑆 ∈ (fBas‘(𝑌 × 𝑌)))
11 ssfg 22484 . . . . . . . . . . . 12 (𝑆 ∈ (fBas‘(𝑌 × 𝑌)) → 𝑆 ⊆ ((𝑌 × 𝑌)filGen𝑆))
1210, 11syl 17 . . . . . . . . . . 11 (𝜑𝑆 ⊆ ((𝑌 × 𝑌)filGen𝑆))
13 isucn2.v . . . . . . . . . . 11 𝑉 = ((𝑌 × 𝑌)filGen𝑆)
1412, 13sseqtrrdi 3966 . . . . . . . . . 10 (𝜑𝑆𝑉)
1514adantr 484 . . . . . . . . 9 ((𝜑𝐹:𝑋𝑌) → 𝑆𝑉)
1615adantr 484 . . . . . . . 8 (((𝜑𝐹:𝑋𝑌) ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) → 𝑆𝑉)
1716sselda 3915 . . . . . . 7 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) ∧ 𝑠𝑆) → 𝑠𝑉)
188, 9, 17rspcdva 3573 . . . . . 6 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) ∧ 𝑠𝑆) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))
19 simpr 488 . . . . . . . . . . . 12 ((𝜑𝑢𝑈) → 𝑢𝑈)
20 isucn2.u . . . . . . . . . . . 12 𝑈 = ((𝑋 × 𝑋)filGen𝑅)
2119, 20eleqtrdi 2900 . . . . . . . . . . 11 ((𝜑𝑢𝑈) → 𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅))
22 isucn2.3 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ (fBas‘(𝑋 × 𝑋)))
23 elfg 22483 . . . . . . . . . . . . 13 (𝑅 ∈ (fBas‘(𝑋 × 𝑋)) → (𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅) ↔ (𝑢 ⊆ (𝑋 × 𝑋) ∧ ∃𝑟𝑅 𝑟𝑢)))
2422, 23syl 17 . . . . . . . . . . . 12 (𝜑 → (𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅) ↔ (𝑢 ⊆ (𝑋 × 𝑋) ∧ ∃𝑟𝑅 𝑟𝑢)))
2524simplbda 503 . . . . . . . . . . 11 ((𝜑𝑢 ∈ ((𝑋 × 𝑋)filGen𝑅)) → ∃𝑟𝑅 𝑟𝑢)
2621, 25syldan 594 . . . . . . . . . 10 ((𝜑𝑢𝑈) → ∃𝑟𝑅 𝑟𝑢)
27 ssbr 5074 . . . . . . . . . . . . . . . . . 18 (𝑟𝑢 → (𝑥𝑟𝑦𝑥𝑢𝑦))
2827imim1d 82 . . . . . . . . . . . . . . . . 17 (𝑟𝑢 → ((𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
2928adantl 485 . . . . . . . . . . . . . . . 16 (((𝜑𝑟𝑅) ∧ 𝑟𝑢) → ((𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
3029ralrimivw 3150 . . . . . . . . . . . . . . 15 (((𝜑𝑟𝑅) ∧ 𝑟𝑢) → ∀𝑦𝑋 ((𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
3130ralrimivw 3150 . . . . . . . . . . . . . 14 (((𝜑𝑟𝑅) ∧ 𝑟𝑢) → ∀𝑥𝑋𝑦𝑋 ((𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
32 ralim 3130 . . . . . . . . . . . . . . 15 (∀𝑦𝑋 ((𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) → (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
3332ralimi 3128 . . . . . . . . . . . . . 14 (∀𝑥𝑋𝑦𝑋 ((𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) → ∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
34 ralim 3130 . . . . . . . . . . . . . 14 (∀𝑥𝑋 (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
3531, 33, 343syl 18 . . . . . . . . . . . . 13 (((𝜑𝑟𝑅) ∧ 𝑟𝑢) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
3635ex 416 . . . . . . . . . . . 12 ((𝜑𝑟𝑅) → (𝑟𝑢 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
3736reximdva 3233 . . . . . . . . . . 11 (𝜑 → (∃𝑟𝑅 𝑟𝑢 → ∃𝑟𝑅 (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
3837adantr 484 . . . . . . . . . 10 ((𝜑𝑢𝑈) → (∃𝑟𝑅 𝑟𝑢 → ∃𝑟𝑅 (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
3926, 38mpd 15 . . . . . . . . 9 ((𝜑𝑢𝑈) → ∃𝑟𝑅 (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
40 r19.37v 3298 . . . . . . . . 9 (∃𝑟𝑅 (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
4139, 40syl 17 . . . . . . . 8 ((𝜑𝑢𝑈) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
4241rexlimdva 3243 . . . . . . 7 (𝜑 → (∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
4342ad3antrrr 729 . . . . . 6 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) ∧ 𝑠𝑆) → (∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
4418, 43mpd 15 . . . . 5 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) ∧ 𝑠𝑆) → ∃𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))
4544ralrimiva 3149 . . . 4 (((𝜑𝐹:𝑋𝑌) ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) → ∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))
46 ssfg 22484 . . . . . . . . . . 11 (𝑅 ∈ (fBas‘(𝑋 × 𝑋)) → 𝑅 ⊆ ((𝑋 × 𝑋)filGen𝑅))
4722, 46syl 17 . . . . . . . . . 10 (𝜑𝑅 ⊆ ((𝑋 × 𝑋)filGen𝑅))
4847, 20sseqtrrdi 3966 . . . . . . . . 9 (𝜑𝑅𝑈)
49 ssrexv 3982 . . . . . . . . . 10 (𝑅𝑈 → (∃𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
50 breq 5032 . . . . . . . . . . . . 13 (𝑟 = 𝑢 → (𝑥𝑟𝑦𝑥𝑢𝑦))
5150imbi1d 345 . . . . . . . . . . . 12 (𝑟 = 𝑢 → ((𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) ↔ (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
52512ralbidv 3164 . . . . . . . . . . 11 (𝑟 = 𝑢 → (∀𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
5352cbvrexvw 3397 . . . . . . . . . 10 (∃𝑟𝑈𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) ↔ ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))
5449, 53syl6ib 254 . . . . . . . . 9 (𝑅𝑈 → (∃𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
5548, 54syl 17 . . . . . . . 8 (𝜑 → (∃𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
5655ralimdv 3145 . . . . . . 7 (𝜑 → (∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
5756adantr 484 . . . . . 6 ((𝜑𝐹:𝑋𝑌) → (∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
58 nfv 1915 . . . . . . . . . . 11 𝑠(𝜑𝐹:𝑋𝑌)
59 nfra1 3183 . . . . . . . . . . 11 𝑠𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))
6058, 59nfan 1900 . . . . . . . . . 10 𝑠((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))
61 nfv 1915 . . . . . . . . . 10 𝑠 𝑣𝑉
6260, 61nfan 1900 . . . . . . . . 9 𝑠(((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉)
63 rspa 3171 . . . . . . . . . . 11 ((∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) ∧ 𝑠𝑆) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))
6463ad5ant24 760 . . . . . . . . . 10 ((((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))
65 simp-4l 782 . . . . . . . . . . 11 ((((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → (𝜑𝐹:𝑋𝑌))
66 simplr 768 . . . . . . . . . . 11 ((((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → 𝑠𝑆)
67 simpr 488 . . . . . . . . . . 11 ((((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → 𝑠𝑣)
68 ssbr 5074 . . . . . . . . . . . . . . . 16 (𝑠𝑣 → ((𝐹𝑥)𝑠(𝐹𝑦) → (𝐹𝑥)𝑣(𝐹𝑦)))
6968adantl 485 . . . . . . . . . . . . . . 15 ((((𝜑𝐹:𝑋𝑌) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → ((𝐹𝑥)𝑠(𝐹𝑦) → (𝐹𝑥)𝑣(𝐹𝑦)))
7069imim2d 57 . . . . . . . . . . . . . 14 ((((𝜑𝐹:𝑋𝑌) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → ((𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))))
7170ralimdv 3145 . . . . . . . . . . . . 13 ((((𝜑𝐹:𝑋𝑌) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → (∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))))
7271ralimdv 3145 . . . . . . . . . . . 12 ((((𝜑𝐹:𝑋𝑌) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))))
7372reximdv 3232 . . . . . . . . . . 11 ((((𝜑𝐹:𝑋𝑌) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → (∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))))
7465, 66, 67, 73syl21anc 836 . . . . . . . . . 10 ((((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → (∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))))
7564, 74mpd 15 . . . . . . . . 9 ((((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) ∧ 𝑠𝑆) ∧ 𝑠𝑣) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))
7610ad3antrrr 729 . . . . . . . . . 10 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) → 𝑆 ∈ (fBas‘(𝑌 × 𝑌)))
77 simpr 488 . . . . . . . . . . 11 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) → 𝑣𝑉)
7877, 13eleqtrdi 2900 . . . . . . . . . 10 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) → 𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆))
79 elfg 22483 . . . . . . . . . . 11 (𝑆 ∈ (fBas‘(𝑌 × 𝑌)) → (𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆) ↔ (𝑣 ⊆ (𝑌 × 𝑌) ∧ ∃𝑠𝑆 𝑠𝑣)))
8079simplbda 503 . . . . . . . . . 10 ((𝑆 ∈ (fBas‘(𝑌 × 𝑌)) ∧ 𝑣 ∈ ((𝑌 × 𝑌)filGen𝑆)) → ∃𝑠𝑆 𝑠𝑣)
8176, 78, 80syl2anc 587 . . . . . . . . 9 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) → ∃𝑠𝑆 𝑠𝑣)
8262, 75, 81r19.29af 3289 . . . . . . . 8 ((((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) ∧ 𝑣𝑉) → ∃𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))
8382ralrimiva 3149 . . . . . . 7 (((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) → ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))
8483ex 416 . . . . . 6 ((𝜑𝐹:𝑋𝑌) → (∀𝑠𝑆𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))))
8557, 84syld 47 . . . . 5 ((𝜑𝐹:𝑋𝑌) → (∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)) → ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))))
8685imp 410 . . . 4 (((𝜑𝐹:𝑋𝑌) ∧ ∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))) → ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)))
8745, 86impbida 800 . . 3 ((𝜑𝐹:𝑋𝑌) → (∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦)) ↔ ∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦))))
8887pm5.32da 582 . 2 (𝜑 → ((𝐹:𝑋𝑌 ∧ ∀𝑣𝑉𝑢𝑈𝑥𝑋𝑦𝑋 (𝑥𝑢𝑦 → (𝐹𝑥)𝑣(𝐹𝑦))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
894, 88bitrd 282 1 (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑠𝑆𝑟𝑅𝑥𝑋𝑦𝑋 (𝑥𝑟𝑦 → (𝐹𝑥)𝑠(𝐹𝑦)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ⊆ wss 3881   class class class wbr 5030   × cxp 5517  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  fBascfbas 20082  filGencfg 20083  UnifOncust 22812   Cnucucn 22888 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8393  df-fbas 20091  df-fg 20092  df-ust 22813  df-ucn 22889 This theorem is referenced by:  metucn  23185
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