Theorem flfcnp2 24031

Description: The image of a convergent sequence under a continuous map is convergent to the image of the original point. Binary operation version. (Contributed by Mario Carneiro, 19-Sep-2015.)

Hypotheses

Ref	Expression
flfcnp2.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
flfcnp2.k	⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
flfcnp2.l	⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍))
flfcnp2.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑋)
flfcnp2.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ 𝑌)
flfcnp2.r	⊢ (𝜑 → 𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐴)))
flfcnp2.s	⊢ (𝜑 → 𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐵)))
flfcnp2.o	⊢ (𝜑 → 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))

Assertion

Ref	Expression
flfcnp2	⊢ (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))

Distinct variable groups:   𝑥,𝑂   𝜑,𝑥   𝑥,𝑍   𝑥,𝑋   𝑥,𝑌

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)   𝑅(𝑥)   𝑆(𝑥)   𝐽(𝑥)   𝐾(𝑥)   𝐿(𝑥)   𝑁(𝑥)

Proof of Theorem flfcnp2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7434	. 2 ⊢ (𝑅𝑂𝑆) = (𝑂‘⟨𝑅, 𝑆⟩)
2		flfcnp2.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		flfcnp2.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		txtopon 23615	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
6		flfcnp2.l	. . . 4 ⊢ (𝜑 → 𝐿 ∈ (Fil‘𝑍))
7		flfcnp2.a	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐴 ∈ 𝑋)
8		flfcnp2.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝐵 ∈ 𝑌)
9	7, 8	opelxpd 5728	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
10	9	fmpttd 7135	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩):𝑍⟶(𝑋 × 𝑌))
11		flfcnp2.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐴)))
12		flfcnp2.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐵)))
13	7	fmpttd 7135	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴):𝑍⟶𝑋)
14	8	fmpttd 7135	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐵):𝑍⟶𝑌)
15		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑦⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩
16		nffvmpt1 6918	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦)
17		nffvmpt1 6918	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)
18	16, 17	nfop 4894	. . . . . . . 8 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)⟩
19		fveq2 6907	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦))
20		fveq2 6907	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦))
21	19, 20	opeq12d 4886	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)⟩)
22	15, 18, 21	cbvmpt 5259	. . . . . . 7 ⊢ (𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩) = (𝑦 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)⟩)
23	2, 3, 6, 13, 14, 22	txflf 24030	. . . . . 6 ⊢ (𝜑 → (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩)) ↔ (𝑅 ∈ ((𝐽 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐴)) ∧ 𝑆 ∈ ((𝐾 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ 𝐵)))))
24	11, 12, 23	mpbir2and 713	. . . . 5 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩)))
25		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍)
26		eqid 2735	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) = (𝑥 ∈ 𝑍 ↦ 𝐴)
27	26	fvmpt2 7027	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
28	25, 7, 27	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
29		eqid 2735	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐵) = (𝑥 ∈ 𝑍 ↦ 𝐵)
30	29	fvmpt2 7027	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝐵 ∈ 𝑌) → ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥) = 𝐵)
31	25, 8, 30	syl2anc 584	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥) = 𝐵)
32	28, 31	opeq12d 4886	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
33	32	mpteq2dva 5248	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))
34	33	fveq2d 6911	. . . . 5 ⊢ (𝜑 → (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩)) = (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩)))
35	24, 34	eleqtrd 2841	. . . 4 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩)))
36		flfcnp2.o	. . . 4 ⊢ (𝜑 → 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))
37		flfcnp 24028	. . . 4 ⊢ ((((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝐿 ∈ (Fil‘𝑍) ∧ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩):𝑍⟶(𝑋 × 𝑌)) ∧ (⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩)) ∧ 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩))) → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))))
38	5, 6, 10, 35, 36, 37	syl32anc 1377	. . 3 ⊢ (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))))
39		eqidd 2736	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))
40		cnptop2 23267	. . . . . . . . 9 ⊢ (𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩) → 𝑁 ∈ Top)
41	36, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Top)
42		toptopon2 22940	. . . . . . . 8 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
43	41, 42	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
44		cnpf2 23274	. . . . . . 7 ⊢ (((𝐽 ×t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)) ∧ 𝑁 ∈ (TopOn‘∪ 𝑁) ∧ 𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩)) → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁)
45	5, 43, 36, 44	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁)
46	45	feqmptd 6977	. . . . 5 ⊢ (𝜑 → 𝑂 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑂‘𝑦)))
47		fveq2 6907	. . . . . 6 ⊢ (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂‘𝑦) = (𝑂‘⟨𝐴, 𝐵⟩))
48		df-ov 7434	. . . . . 6 ⊢ (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
49	47, 48	eqtr4di 2793	. . . . 5 ⊢ (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂‘𝑦) = (𝐴𝑂𝐵))
50	9, 39, 46, 49	fmptco 7149	. . . 4 ⊢ (𝜑 → (𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩)) = (𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵)))
51	50	fveq2d 6911	. . 3 ⊢ (𝜑 → ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))) = ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))
52	38, 51	eleqtrd 2841	. 2 ⊢ (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))
53	1, 52	eqeltrid 2843	1 ⊢ (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ⟨cop 4637  ∪ cuni 4912   ↦ cmpt 5231   × cxp 5687   ∘ ccom 5693  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  Topctop 22915  TopOnctopon 22932   CnP ccnp 23249   ×_t ctx 23584  Filcfil 23869   fLimf cflf 23959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-topgen 17490  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-bases 22969  df-ntr 23044  df-nei 23122  df-cnp 23252  df-tx 23586  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964

This theorem is referenced by:  tsmsadd  24171