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	⊢ (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 7316	. . . 4 ⊢ (𝐴𝐹𝐵) = (𝐹‘⟨𝐴, 𝐵⟩)
4		txcnpi.7	. . . 4 ⊢ (𝜑 → (𝐴𝐹𝐵) ∈ 𝑈)
5	3, 4	eqeltrrid 2843	. . 3 ⊢ (𝜑 → (𝐹‘⟨𝐴, 𝐵⟩) ∈ 𝑈)
6		cnpimaex 22478	. . 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 1370	. 2 ⊢ (𝜑 → ∃𝑤 ∈ (𝐽 ×t 𝐾)(⟨𝐴, 𝐵⟩ ∈ 𝑤 ∧ (𝐹 “ 𝑤) ⊆ 𝑈))
8		eqid 2737	. . . . . . . . . 10 ⊢ ∪ (𝐽 ×t 𝐾) = ∪ (𝐽 ×t 𝐾)
9		eqid 2737	. . . . . . . . . 10 ⊢ ∪ 𝐿 = ∪ 𝐿
10	8, 9	cnpf 22469	. . . . . . . . 9 ⊢ (𝐹 ∈ (((𝐽 ×t 𝐾) CnP 𝐿)‘⟨𝐴, 𝐵⟩) → 𝐹:∪ (𝐽 ×t 𝐾)⟶∪ 𝐿)
11	1, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐹:∪ (𝐽 ×t 𝐾)⟶∪ 𝐿)
12	11	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → 𝐹:∪ (𝐽 ×t 𝐾)⟶∪ 𝐿)
13	12	ffund 6639	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → Fun 𝐹)
14		elssuni 4881	. . . . . . 7 ⊢ (𝑤 ∈ (𝐽 ×t 𝐾) → 𝑤 ⊆ ∪ (𝐽 ×t 𝐾))
15	11	fdmd 6646	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = ∪ (𝐽 ×t 𝐾))
16	15	sseq2d 3962	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ⊆ dom 𝐹 ↔ 𝑤 ⊆ ∪ (𝐽 ×t 𝐾)))
17	16	biimpar 478	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ⊆ ∪ (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
18	14, 17	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → 𝑤 ⊆ dom 𝐹)
19		funimass3 6968	. . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ 𝑈 ↔ 𝑤 ⊆ (◡𝐹 “ 𝑈)))
20	13, 18, 19	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → ((𝐹 “ 𝑤) ⊆ 𝑈 ↔ 𝑤 ⊆ (◡𝐹 “ 𝑈)))
21	20	anbi2d 629	. . . 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 22790	. . . . . . 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 584	. . . . . 6 ⊢ (𝜑 → (𝑤 ∈ (𝐽 ×t 𝐾) ↔ ∀𝑧 ∈ 𝑤 ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
26	25	biimpa 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐽 ×t 𝐾)) → ∀𝑧 ∈ 𝑤 ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
27		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝐴, 𝐵⟩ → (𝑧 ∈ (𝑢 × 𝑣) ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣)))
28	27	anbi1d 630	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝐴, 𝐵⟩ → ((𝑧 ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤)))
29	28	2rexbidv 3210	. . . . . . . 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 3567	. . . . . . 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 3937	. . . . . . . . . . . . 13 ⊢ ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑤 ⊆ (◡𝐹 “ 𝑈) → (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))
32	31	com12 32	. . . . . . . . . . . 12 ⊢ (𝑤 ⊆ (◡𝐹 “ 𝑈) → ((𝑢 × 𝑣) ⊆ 𝑤 → (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))
33	32	anim2d 612	. . . . . . . . . . 11 ⊢ (𝑤 ⊆ (◡𝐹 “ 𝑈) → (((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → ((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
34		opelxp 5641	. . . . . . . . . . . 12 ⊢ (⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ↔ (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣))
35	34	anbi1i 624	. . . . . . . . . . 11 ⊢ ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) ↔ ((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤))
36		df-3an 1088	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)) ↔ ((𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣) ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))
37	33, 35, 36	3imtr4g 295	. . . . . . . . . 10 ⊢ (𝑤 ⊆ (◡𝐹 “ 𝑈) → ((⟨𝐴, 𝐵⟩ ∈ (𝑢 × 𝑣) ∧ (𝑢 × 𝑣) ⊆ 𝑤) → (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈))))
38	37	reximdv 3164	. . . . . . . . 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 3164	. . . . . . . 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 32	. . . . . . 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 35	. . . . . 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 411	. . . . 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 17	. . . 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 239	. . 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 3149	. 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 15	1 ⊢ (𝜑 → ∃𝑢 ∈ 𝐽 ∃𝑣 ∈ 𝐾 (𝐴 ∈ 𝑢 ∧ 𝐵 ∈ 𝑣 ∧ (𝑢 × 𝑣) ⊆ (◡𝐹 “ 𝑈)))

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