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Theorem txcnpi 22830
Description: Continuity of a two-argument function at a point. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
txcnpi.1 (𝜑𝐽 ∈ (TopOn‘𝑋))
txcnpi.2 (𝜑𝐾 ∈ (TopOn‘𝑌))
txcnpi.3 (𝜑𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))
txcnpi.4 (𝜑𝑈𝐿)
txcnpi.5 (𝜑𝐴𝑋)
txcnpi.6 (𝜑𝐵𝑌)
txcnpi.7 (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)
Assertion
Ref Expression
txcnpi (𝜑 → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
Distinct variable groups:   𝑣,𝑢,𝐴   𝑢,𝐵,𝑣   𝑢,𝐹,𝑣   𝑢,𝐽,𝑣   𝑢,𝐾,𝑣   𝑢,𝑈,𝑣
Allowed substitution hints:   𝜑(𝑣,𝑢)   𝐿(𝑣,𝑢)   𝑋(𝑣,𝑢)   𝑌(𝑣,𝑢)

Proof of Theorem txcnpi
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 txcnpi.3 . . 3 (𝜑𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))
2 txcnpi.4 . . 3 (𝜑𝑈𝐿)
3 df-ov 7316 . . . 4 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4 txcnpi.7 . . . 4 (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)
53, 4eqeltrrid 2843 . . 3 (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝑈)
6 cnpimaex 22478 . . 3 ((𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ∧ 𝑈𝐿 ∧ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝑈) → ∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈))
71, 2, 5, 6syl3anc 1370 . 2 (𝜑 → ∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈))
8 eqid 2737 . . . . . . . . . 10 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
9 eqid 2737 . . . . . . . . . 10 𝐿 = 𝐿
108, 9cnpf 22469 . . . . . . . . 9 (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) → 𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
111, 10syl 17 . . . . . . . 8 (𝜑𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
1211adantr 481 . . . . . . 7 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → 𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
1312ffund 6639 . . . . . 6 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → Fun 𝐹)
14 elssuni 4881 . . . . . . 7 (𝑤 ∈ (𝐽 ×t 𝐾) → 𝑤 (𝐽 ×t 𝐾))
1511fdmd 6646 . . . . . . . . 9 (𝜑 → dom 𝐹 = (𝐽 ×t 𝐾))
1615sseq2d 3962 . . . . . . . 8 (𝜑 → (𝑤 ⊆ dom 𝐹𝑤 (𝐽 ×t 𝐾)))
1716biimpar 478 . . . . . . 7 ((𝜑𝑤 (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
1814, 17sylan2 593 . . . . . 6 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
19 funimass3 6968 . . . . . 6 ((Fun 𝐹𝑤 ⊆ dom 𝐹) → ((𝐹𝑤) ⊆ 𝑈𝑤 ⊆ (𝐹𝑈)))
2013, 18, 19syl2anc 584 . . . . 5 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((𝐹𝑤) ⊆ 𝑈𝑤 ⊆ (𝐹𝑈)))
2120anbi2d 629 . . . 4 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑤𝑤 ⊆ (𝐹𝑈))))
22 txcnpi.1 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
23 txcnpi.2 . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑌))
24 eltx 22790 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2522, 23, 24syl2anc 584 . . . . . 6 (𝜑 → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2625biimpa 477 . . . . 5 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
27 eleq1 2825 . . . . . . . . . 10 (𝑧 = ⟨𝐴, 𝐵⟩ → (𝑧 ∈ (𝑢 × 𝑣) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣)))
2827anbi1d 630 . . . . . . . . 9 (𝑧 = ⟨𝐴, 𝐵⟩ → ((𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
29282rexbidv 3210 . . . . . . . 8 (𝑧 = ⟨𝐴, 𝐵⟩ → (∃𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
3029rspccv 3567 . . . . . . 7 (∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (⟨𝐴, 𝐵⟩ ∈ 𝑤 → ∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
31 sstr2 3937 . . . . . . . . . . . . 13 ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑤 ⊆ (𝐹𝑈) → (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3231com12 32 . . . . . . . . . . . 12 (𝑤 ⊆ (𝐹𝑈) → ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3332anim2d 612 . . . . . . . . . . 11 (𝑤 ⊆ (𝐹𝑈) → (((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
34 opelxp 5641 . . . . . . . . . . . 12 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ↔ (𝐴𝑢𝐵𝑣))
3534anbi1i 624 . . . . . . . . . . 11 ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
36 df-3an 1088 . . . . . . . . . . 11 ((𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)) ↔ ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3733, 35, 363imtr4g 295 . . . . . . . . . 10 (𝑤 ⊆ (𝐹𝑈) → ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
3837reximdv 3164 . . . . . . . . 9 (𝑤 ⊆ (𝐹𝑈) → (∃𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
3938reximdv 3164 . . . . . . . 8 (𝑤 ⊆ (𝐹𝑈) → (∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4039com12 32 . . . . . . 7 (∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝑤 ⊆ (𝐹𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4130, 40syl6 35 . . . . . 6 (∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (⟨𝐴, 𝐵⟩ ∈ 𝑤 → (𝑤 ⊆ (𝐹𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))))
4241impd 411 . . . . 5 (∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤𝑤 ⊆ (𝐹𝑈)) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4326, 42syl 17 . . . 4 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤𝑤 ⊆ (𝐹𝑈)) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4421, 43sylbid 239 . . 3 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4544rexlimdva 3149 . 2 (𝜑 → (∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
467, 45mpd 15 1 (𝜑 → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1540  wcel 2105  wral 3062  wrex 3071  wss 3896  cop 4575   cuni 4848   × cxp 5603  ccnv 5604  dom cdm 5605  cima 5608  Fun wfun 6457  wf 6459  cfv 6463  (class class class)co 7313  TopOnctopon 22130   CnP ccnp 22447   ×t ctx 22782
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5236  ax-nul 5243  ax-pow 5301  ax-pr 5365  ax-un 7626
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3443  df-sbc 3726  df-csb 3842  df-dif 3899  df-un 3901  df-in 3903  df-ss 3913  df-nul 4267  df-if 4470  df-pw 4545  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4849  df-iun 4937  df-br 5086  df-opab 5148  df-mpt 5169  df-id 5505  df-xp 5611  df-rel 5612  df-cnv 5613  df-co 5614  df-dm 5615  df-rn 5616  df-res 5617  df-ima 5618  df-iota 6415  df-fun 6465  df-fn 6466  df-f 6467  df-fv 6471  df-ov 7316  df-oprab 7317  df-mpo 7318  df-1st 7874  df-2nd 7875  df-map 8663  df-topgen 17221  df-top 22114  df-topon 22131  df-cnp 22450  df-tx 22784
This theorem is referenced by:  tmdcn2  23311
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