Theorem txcnpi 23744

Description: Continuity of a two-argument function at a point. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
txcnpi.1	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
txcnpi.2	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
txcnpi.3	⊢ (𝜑 → 𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))
txcnpi.4	⊢ (𝜑 → 𝑈 ∈ 𝐿)
txcnpi.5	⊢ (𝜑 → 𝐴 ∈ 𝑋)
txcnpi.6	⊢ (𝜑 → 𝐵 ∈ 𝑌)
txcnpi.7	⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)

Assertion

Ref	Expression
txcnpi	⊢ (𝜑 → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))

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.

Step	Hyp	Ref	Expression
1		txcnpi.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩))
2		txcnpi.4	. . 3 ⊢ (𝜑 → 𝑈 ∈ 𝐿)
3		df-ov 7413	. . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4		txcnpi.7	. . . 4 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)
5	3, 4	eqeltrrid 2866	. . 3 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝑈)
6		cnpimaex 23392	. . 3 ⊢ ((𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) ∧ 𝑈 ∈ 𝐿 ∧ (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝑈) → ∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹 “ 𝑤) ⊆ 𝑈))
7	1, 2, 5, 6	syl3anc 1396	. 2 ⊢ (𝜑 → ∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹 “ 𝑤) ⊆ 𝑈))
8		eqid 2761	. . . . . . . . . 10 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
9		eqid 2761	. . . . . . . . . 10 ⊢ ∪ 𝐿 = ∪ 𝐿
10	8, 9	cnpf 23383	. . . . . . . . 9 ⊢ (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) → 𝐹:∪ (𝐽 ×t 𝐾)⟶∪ 𝐿)
11	1, 10	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐹:∪ (𝐽 ×t 𝐾)⟶∪ 𝐿)
12	11	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → 𝐹:∪ (𝐽 ×t 𝐾)⟶∪ 𝐿)
13	12	ffund 6710	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → Fun 𝐹)
14		elssuni 4903	. . . . . . 7 ⊢ (𝑤 ∈ (𝐽 ×t 𝐾) → 𝑤 ⊆ ∪ (𝐽 ×t 𝐾))
15	11	fdmd 6716	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = ∪ (𝐽 ×t 𝐾))
16	15	sseq2d 3968	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ⊆ dom 𝐹 ↔ 𝑤 ⊆ ∪ (𝐽 ×t 𝐾)))
17	16	biimpar 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ⊆ ∪ (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
18	14, 17	sylan2 604	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
19		funimass3 7049	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ 𝑈 ↔ 𝑤 ⊆ (◡𝐹 “ 𝑈)))
20	13, 18, 19	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → ((𝐹 “ 𝑤) ⊆ 𝑈 ↔ 𝑤 ⊆ (◡𝐹 “ 𝑈)))
21	20	anbi2d 641	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹 “ 𝑤) ⊆ 𝑈) ↔ (⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ 𝑤 ⊆ (◡𝐹 “ 𝑈))))
22		txcnpi.1	. . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
23		txcnpi.2	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
24		eltx 23704	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝑤 ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
25	22, 23, 24	syl2anc 595	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝑤 ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
26	25	biimpa 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → ∀𝑧 ∈ 𝑤 ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
27		eleq1 2849	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝐴, 𝐵⟩ → (𝑧 ∈ (𝑢 × 𝑣) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣)))
28	27	anbi1d 642	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝐴, 𝐵⟩ → ((𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
29	28	2rexbidv 3228	. . . . . . . 8 ⊢ (𝑧 = ⟨𝐴, 𝐵⟩ → (∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
30	29	rspccv 3577	. . . . . . 7 ⊢ (∀𝑧 ∈ 𝑤 ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (⟨𝐴, 𝐵⟩ ∈ 𝑤 → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
31		sstr2 3943	. . . . . . . . . . . . 13 ⊢ ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑤 ⊆ (◡𝐹 “ 𝑈) → (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))
32	31	com12 33	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ (◡𝐹 “ 𝑈) → ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))
33	32	anim2d 623	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ (◡𝐹 “ 𝑈) → (((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
34		opelxp 5697	. . . . . . . . . . . 12 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ↔ (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣))
35	34	anbi1i 635	. . . . . . . . . . 11 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
36		df-3an 1103	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)) ↔ ((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))
37	33, 35, 36	3imtr4g 299	. . . . . . . . . 10 ⊢ (𝑤 ⊆ (◡𝐹 “ 𝑈) → ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
38	37	reximdv 3178	. . . . . . . . 9 ⊢ (𝑤 ⊆ (◡𝐹 “ 𝑈) → (∃𝑣 ∈ 𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
39	38	reximdv 3178	. . . . . . . 8 ⊢ (𝑤 ⊆ (◡𝐹 “ 𝑈) → (∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
40	39	com12 33	. . . . . . 7 ⊢ (∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝑤 ⊆ (◡𝐹 “ 𝑈) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
41	30, 40	syl6 36	. . . . . 6 ⊢ (∀𝑧 ∈ 𝑤 ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (⟨𝐴, 𝐵⟩ ∈ 𝑤 → (𝑤 ⊆ (◡𝐹 “ 𝑈) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))))
42	41	impd 415	. . . . 5 ⊢ (∀𝑧 ∈ 𝑤 ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ 𝑤 ⊆ (◡𝐹 “ 𝑈)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
43	26, 42	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ 𝑤 ⊆ (◡𝐹 “ 𝑈)) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
44	21, 43	sylbid 243	. . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → ((⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹 “ 𝑤) ⊆ 𝑈) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
45	44	rexlimdva 3164	. 2 ⊢ (𝜑 → (∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹 “ 𝑤) ⊆ 𝑈) → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
46	7, 45	mpd 16	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087   ⊆ wss 3904  ⟨cop 4594  ∪ cuni 4871   × cxp 5659  ^◡ccnv 5660  dom cdm 5661   “ cima 5664  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  TopOnctopon 23046   CnP ccnp 23361   ×_t ctx 23696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7985  df-2nd 7986  df-map 8825  df-topgen 17495  df-top 23030  df-topon 23047  df-cnp 23364  df-tx 23698

This theorem is referenced by:  tmdcn2  24225