Theorem flfcnp2 23395

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 7365	. 2 ⊢ (𝑅𝑂𝑆) = (𝑂‘⟨𝑅, 𝑆⟩)
2		flfcnp2.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		flfcnp2.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		txtopon 22979	. . . . 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 5676	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
10	9	fmpttd 7068	. . . 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 7068	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴):𝑍⟶𝑋)
14	8	fmpttd 7068	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐵):𝑍⟶𝑌)
15		nfcv 2902	. . . . . . . 8 ⊢ Ⅎ𝑦⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩
16		nffvmpt1 6858	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦)
17		nffvmpt1 6858	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)
18	16, 17	nfop 4851	. . . . . . . 8 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)⟩
19		fveq2 6847	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦))
20		fveq2 6847	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦))
21	19, 20	opeq12d 4843	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)⟩)
22	15, 18, 21	cbvmpt 5221	. . . . . . 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 23394	. . . . . 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 711	. . . . 5 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×t 𝐾) fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩)))
25		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍)
26		eqid 2731	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) = (𝑥 ∈ 𝑍 ↦ 𝐴)
27	26	fvmpt2 6964	. . . . . . . . 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 2731	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐵) = (𝑥 ∈ 𝑍 ↦ 𝐵)
30	29	fvmpt2 6964	. . . . . . . . 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 4843	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
33	32	mpteq2dva 5210	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))
34	33	fveq2d 6851	. . . . 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 2834	. . . 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 23392	. . . 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 1378	. . 3 ⊢ (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))))
39		eqidd 2732	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))
40		cnptop2 22631	. . . . . . . . 9 ⊢ (𝑂 ∈ (((𝐽 ×t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩) → 𝑁 ∈ Top)
41	36, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Top)
42		toptopon2 22304	. . . . . . . 8 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
43	41, 42	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
44		cnpf2 22638	. . . . . . 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 1371	. . . . . 6 ⊢ (𝜑 → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁)
46	45	feqmptd 6915	. . . . 5 ⊢ (𝜑 → 𝑂 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑂‘𝑦)))
47		fveq2 6847	. . . . . 6 ⊢ (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂‘𝑦) = (𝑂‘⟨𝐴, 𝐵⟩))
48		df-ov 7365	. . . . . 6 ⊢ (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
49	47, 48	eqtr4di 2789	. . . . 5 ⊢ (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂‘𝑦) = (𝐴𝑂𝐵))
50	9, 39, 46, 49	fmptco 7080	. . . 4 ⊢ (𝜑 → (𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩)) = (𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵)))
51	50	fveq2d 6851	. . 3 ⊢ (𝜑 → ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))) = ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))
52	38, 51	eleqtrd 2834	. 2 ⊢ (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))
53	1, 52	eqeltrid 2836	1 ⊢ (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ⟨cop 4597  ∪ cuni 4870   ↦ cmpt 5193   × cxp 5636   ∘ ccom 5642  ⟶wf 6497  ‘cfv 6501  (class class class)co 7362  Topctop 22279  TopOnctopon 22296   CnP ccnp 22613   ×_t ctx 22948  Filcfil 23233   fLimf cflf 23323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-oprab 7366  df-mpo 7367  df-1st 7926  df-2nd 7927  df-map 8774  df-topgen 17339  df-fbas 20830  df-fg 20831  df-top 22280  df-topon 22297  df-bases 22333  df-ntr 22408  df-nei 22486  df-cnp 22616  df-tx 22950  df-fil 23234  df-fm 23326  df-flim 23327  df-flf 23328

This theorem is referenced by:  tsmsadd  23535