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Theorem txcnpi 22144
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 7148 . . . 4 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4 txcnpi.7 . . . 4 (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)
53, 4eqeltrrid 2915 . . 3 (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝑈)
6 cnpimaex 21792 . . 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 1363 . 2 (𝜑 → ∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈))
8 eqid 2818 . . . . . . . . . 10 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
9 eqid 2818 . . . . . . . . . 10 𝐿 = 𝐿
108, 9cnpf 21783 . . . . . . . . 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 6511 . . . . . 6 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → Fun 𝐹)
14 elssuni 4859 . . . . . . 7 (𝑤 ∈ (𝐽 ×t 𝐾) → 𝑤 (𝐽 ×t 𝐾))
1511fdmd 6516 . . . . . . . . 9 (𝜑 → dom 𝐹 = (𝐽 ×t 𝐾))
1615sseq2d 3996 . . . . . . . 8 (𝜑 → (𝑤 ⊆ dom 𝐹𝑤 (𝐽 ×t 𝐾)))
1716biimpar 478 . . . . . . 7 ((𝜑𝑤 (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
1814, 17sylan2 592 . . . . . 6 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
19 funimass3 6816 . . . . . 6 ((Fun 𝐹𝑤 ⊆ dom 𝐹) → ((𝐹𝑤) ⊆ 𝑈𝑤 ⊆ (𝐹𝑈)))
2013, 18, 19syl2anc 584 . . . . 5 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((𝐹𝑤) ⊆ 𝑈𝑤 ⊆ (𝐹𝑈)))
2120anbi2d 628 . . . 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 22104 . . . . . . 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 2897 . . . . . . . . . 10 (𝑧 = ⟨𝐴, 𝐵⟩ → (𝑧 ∈ (𝑢 × 𝑣) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣)))
2827anbi1d 629 . . . . . . . . 9 (𝑧 = ⟨𝐴, 𝐵⟩ → ((𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
29282rexbidv 3297 . . . . . . . 8 (𝑧 = ⟨𝐴, 𝐵⟩ → (∃𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
3029rspccv 3617 . . . . . . 7 (∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (⟨𝐴, 𝐵⟩ ∈ 𝑤 → ∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
31 sstr2 3971 . . . . . . . . . . . . 13 ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑤 ⊆ (𝐹𝑈) → (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3231com12 32 . . . . . . . . . . . 12 (𝑤 ⊆ (𝐹𝑈) → ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3332anim2d 611 . . . . . . . . . . 11 (𝑤 ⊆ (𝐹𝑈) → (((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
34 opelxp 5584 . . . . . . . . . . . 12 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ↔ (𝐴𝑢𝐵𝑣))
3534anbi1i 623 . . . . . . . . . . 11 ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
36 df-3an 1081 . . . . . . . . . . 11 ((𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)) ↔ ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3733, 35, 363imtr4g 297 . . . . . . . . . 10 (𝑤 ⊆ (𝐹𝑈) → ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
3837reximdv 3270 . . . . . . . . 9 (𝑤 ⊆ (𝐹𝑈) → (∃𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
3938reximdv 3270 . . . . . . . 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 241 . . 3 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4544rexlimdva 3281 . 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  wss 3933  cop 4563   cuni 4830   × cxp 5546  ccnv 5547  dom cdm 5548  cima 5551  Fun wfun 6342  wf 6344  cfv 6348  (class class class)co 7145  TopOnctopon 21446   CnP ccnp 21761   ×t ctx 22096
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-topgen 16705  df-top 21430  df-topon 21447  df-cnp 21764  df-tx 22098
This theorem is referenced by:  tmdcn2  22625
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