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Theorem txcnpi 21691
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 6845 . . . 4 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4 txcnpi.7 . . . 4 (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)
53, 4syl5eqelr 2849 . . 3 (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝑈)
6 cnpimaex 21340 . . 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 1490 . 2 (𝜑 → ∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈))
8 eqid 2765 . . . . . . . . . 10 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
9 eqid 2765 . . . . . . . . . 10 𝐿 = 𝐿
108, 9cnpf 21331 . . . . . . . . 9 (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) → 𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
111, 10syl 17 . . . . . . . 8 (𝜑𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
1211adantr 472 . . . . . . 7 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → 𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
1312ffund 6227 . . . . . 6 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → Fun 𝐹)
14 elssuni 4625 . . . . . . 7 (𝑤 ∈ (𝐽 ×t 𝐾) → 𝑤 (𝐽 ×t 𝐾))
1511fdmd 6232 . . . . . . . . 9 (𝜑 → dom 𝐹 = (𝐽 ×t 𝐾))
1615sseq2d 3793 . . . . . . . 8 (𝜑 → (𝑤 ⊆ dom 𝐹𝑤 (𝐽 ×t 𝐾)))
1716biimpar 469 . . . . . . 7 ((𝜑𝑤 (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
1814, 17sylan2 586 . . . . . 6 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
19 funimass3 6523 . . . . . 6 ((Fun 𝐹𝑤 ⊆ dom 𝐹) → ((𝐹𝑤) ⊆ 𝑈𝑤 ⊆ (𝐹𝑈)))
2013, 18, 19syl2anc 579 . . . . 5 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((𝐹𝑤) ⊆ 𝑈𝑤 ⊆ (𝐹𝑈)))
2120anbi2d 622 . . . 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 21651 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2522, 23, 24syl2anc 579 . . . . . 6 (𝜑 → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2625biimpa 468 . . . . 5 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
27 eleq1 2832 . . . . . . . . . 10 (𝑧 = ⟨𝐴, 𝐵⟩ → (𝑧 ∈ (𝑢 × 𝑣) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣)))
2827anbi1d 623 . . . . . . . . 9 (𝑧 = ⟨𝐴, 𝐵⟩ → ((𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
29282rexbidv 3204 . . . . . . . 8 (𝑧 = ⟨𝐴, 𝐵⟩ → (∃𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
3029rspccv 3458 . . . . . . 7 (∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (⟨𝐴, 𝐵⟩ ∈ 𝑤 → ∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
31 sstr2 3768 . . . . . . . . . . . . 13 ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑤 ⊆ (𝐹𝑈) → (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3231com12 32 . . . . . . . . . . . 12 (𝑤 ⊆ (𝐹𝑈) → ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3332anim2d 605 . . . . . . . . . . 11 (𝑤 ⊆ (𝐹𝑈) → (((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
34 opelxp 5313 . . . . . . . . . . . 12 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ↔ (𝐴𝑢𝐵𝑣))
3534anbi1i 617 . . . . . . . . . . 11 ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
36 df-3an 1109 . . . . . . . . . . 11 ((𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)) ↔ ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3733, 35, 363imtr4g 287 . . . . . . . . . 10 (𝑤 ⊆ (𝐹𝑈) → ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
3837reximdv 3162 . . . . . . . . 9 (𝑤 ⊆ (𝐹𝑈) → (∃𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
3938reximdv 3162 . . . . . . . 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 398 . . . . 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 231 . . 3 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4544rexlimdva 3178 . 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  wss 3732  cop 4340   cuni 4594   × cxp 5275  ccnv 5276  dom cdm 5277  cima 5280  Fun wfun 6062  wf 6064  cfv 6068  (class class class)co 6842  TopOnctopon 20994   CnP ccnp 21309   ×t ctx 21643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062  df-topgen 16372  df-top 20978  df-topon 20995  df-cnp 21312  df-tx 21645
This theorem is referenced by:  tmdcn2  22172
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