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Theorem txcnpi 21459
 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 6693 . . . 4 (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4 txcnpi.7 . . . 4 (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)
53, 4syl5eqelr 2735 . . 3 (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝑈)
6 cnpimaex 21108 . . 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 1366 . 2 (𝜑 → ∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈))
8 eqid 2651 . . . . . . . . . 10 (𝐽 ×t 𝐾) = (𝐽 ×t 𝐾)
9 eqid 2651 . . . . . . . . . 10 𝐿 = 𝐿
108, 9cnpf 21099 . . . . . . . . 9 (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) → 𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
111, 10syl 17 . . . . . . . 8 (𝜑𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
1211adantr 480 . . . . . . 7 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → 𝐹: (𝐽 ×t 𝐾)⟶ 𝐿)
13 ffun 6086 . . . . . . 7 (𝐹: (𝐽 ×t 𝐾)⟶ 𝐿 → Fun 𝐹)
1412, 13syl 17 . . . . . 6 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → Fun 𝐹)
15 elssuni 4499 . . . . . . 7 (𝑤 ∈ (𝐽 ×t 𝐾) → 𝑤 (𝐽 ×t 𝐾))
16 fdm 6089 . . . . . . . . . 10 (𝐹: (𝐽 ×t 𝐾)⟶ 𝐿 → dom 𝐹 = (𝐽 ×t 𝐾))
1711, 16syl 17 . . . . . . . . 9 (𝜑 → dom 𝐹 = (𝐽 ×t 𝐾))
1817sseq2d 3666 . . . . . . . 8 (𝜑 → (𝑤 ⊆ dom 𝐹𝑤 (𝐽 ×t 𝐾)))
1918biimpar 501 . . . . . . 7 ((𝜑𝑤 (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
2015, 19sylan2 490 . . . . . 6 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
21 funimass3 6373 . . . . . 6 ((Fun 𝐹𝑤 ⊆ dom 𝐹) → ((𝐹𝑤) ⊆ 𝑈𝑤 ⊆ (𝐹𝑈)))
2214, 20, 21syl2anc 694 . . . . 5 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((𝐹𝑤) ⊆ 𝑈𝑤 ⊆ (𝐹𝑈)))
2322anbi2d 740 . . . 4 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑤𝑤 ⊆ (𝐹𝑈))))
24 txcnpi.1 . . . . . . 7 (𝜑𝐽 ∈ (TopOn‘𝑋))
25 txcnpi.2 . . . . . . 7 (𝜑𝐾 ∈ (TopOn‘𝑌))
26 eltx 21419 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2724, 25, 26syl2anc 694 . . . . . 6 (𝜑 → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
2827biimpa 500 . . . . 5 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
29 eleq1 2718 . . . . . . . . . 10 (𝑧 = ⟨𝐴, 𝐵⟩ → (𝑧 ∈ (𝑢 × 𝑣) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣)))
3029anbi1d 741 . . . . . . . . 9 (𝑧 = ⟨𝐴, 𝐵⟩ → ((𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
31302rexbidv 3086 . . . . . . . 8 (𝑧 = ⟨𝐴, 𝐵⟩ → (∃𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
3231rspccv 3337 . . . . . . 7 (∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (⟨𝐴, 𝐵⟩ ∈ 𝑤 → ∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
33 sstr2 3643 . . . . . . . . . . . . 13 ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑤 ⊆ (𝐹𝑈) → (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3433com12 32 . . . . . . . . . . . 12 (𝑤 ⊆ (𝐹𝑈) → ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3534anim2d 588 . . . . . . . . . . 11 (𝑤 ⊆ (𝐹𝑈) → (((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
36 opelxp 5180 . . . . . . . . . . . 12 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ↔ (𝐴𝑢𝐵𝑣))
3736anbi1i 731 . . . . . . . . . . 11 ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
38 df-3an 1056 . . . . . . . . . . 11 ((𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)) ↔ ((𝐴𝑢𝐵𝑣) ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
3935, 37, 383imtr4g 285 . . . . . . . . . 10 (𝑤 ⊆ (𝐹𝑈) → ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4039reximdv 3045 . . . . . . . . 9 (𝑤 ⊆ (𝐹𝑈) → (∃𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4140reximdv 3045 . . . . . . . 8 (𝑤 ⊆ (𝐹𝑈) → (∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4241com12 32 . . . . . . 7 (∃𝑢𝐽𝑣𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝑤 ⊆ (𝐹𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4332, 42syl6 35 . . . . . 6 (∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (⟨𝐴, 𝐵⟩ ∈ 𝑤 → (𝑤 ⊆ (𝐹𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))))
4443impd 446 . . . . 5 (∀𝑧𝑤𝑢𝐽𝑣𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤𝑤 ⊆ (𝐹𝑈)) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4528, 44syl 17 . . . 4 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤𝑤 ⊆ (𝐹𝑈)) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4623, 45sylbid 230 . . 3 ((𝜑𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
4746rexlimdva 3060 . 2 (𝜑 → (∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹𝑤) ⊆ 𝑈) → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈))))
487, 47mpd 15 1 (𝜑 → ∃𝑢𝐽𝑣𝐾 (𝐴𝑢𝐵𝑣 ∧ (𝑢 × 𝑣) ⊆ (𝐹𝑈)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ⊆ wss 3607  ⟨cop 4216  ∪ cuni 4468   × cxp 5141  ◡ccnv 5142  dom cdm 5143   “ cima 5146  Fun wfun 5920  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690  TopOnctopon 20763   CnP ccnp 21077   ×t ctx 21411 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901  df-topgen 16151  df-top 20747  df-topon 20764  df-cnp 21080  df-tx 21413 This theorem is referenced by:  tmdcn2  21940
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