flfcnp2 - Metamath Proof Explorer

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Theorem flfcnp2 23511

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 7412	. 2 ⊢ (𝑅𝑂𝑆) = (𝑂‘⟨𝑅, 𝑆⟩)
2		flfcnp2.j	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		flfcnp2.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑌))
4		txtopon 23095	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐽 ×_t 𝐾) ∈ (TopOn‘(𝑋 × 𝑌)))
5	2, 3, 4	syl2anc 585	. . . 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 5716	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ⟨𝐴, 𝐵⟩ ∈ (𝑋 × 𝑌))
10	9	fmpttd 7115	. . . 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 7115	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐴):𝑍⟶𝑋)
14	8	fmpttd 7115	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ 𝐵):𝑍⟶𝑌)
15		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑦⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩
16		nffvmpt1 6903	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦)
17		nffvmpt1 6903	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)
18	16, 17	nfop 4890	. . . . . . . 8 ⊢ Ⅎ𝑥⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)⟩
19		fveq2 6892	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦))
20		fveq2 6892	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦))
21	19, 20	opeq12d 4882	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩ = ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑦), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑦)⟩)
22	15, 18, 21	cbvmpt 5260	. . . . . . 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 23510	. . . . . 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 712	. . . . 5 ⊢ (𝜑 → ⟨𝑅, 𝑆⟩ ∈ (((𝐽 ×_t 𝐾) fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩)))
25		simpr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → 𝑥 ∈ 𝑍)
26		eqid 2733	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐴) = (𝑥 ∈ 𝑍 ↦ 𝐴)
27	26	fvmpt2 7010	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝐴 ∈ 𝑋) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
28	25, 7, 27	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥) = 𝐴)
29		eqid 2733	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑍 ↦ 𝐵) = (𝑥 ∈ 𝑍 ↦ 𝐵)
30	29	fvmpt2 7010	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝑍 ∧ 𝐵 ∈ 𝑌) → ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥) = 𝐵)
31	25, 8, 30	syl2anc 585	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥) = 𝐵)
32	28, 31	opeq12d 4882	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑍) → ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩ = ⟨𝐴, 𝐵⟩)
33	32	mpteq2dva 5249	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ ⟨((𝑥 ∈ 𝑍 ↦ 𝐴)‘𝑥), ((𝑥 ∈ 𝑍 ↦ 𝐵)‘𝑥)⟩) = (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))
34	33	fveq2d 6896	. . . . 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 2836	. . . 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 23508	. . . 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 1379	. . 3 ⊢ (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))))
39		eqidd 2734	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩) = (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))
40		cnptop2 22747	. . . . . . . . 9 ⊢ (𝑂 ∈ (((𝐽 ×_t 𝐾) CnP 𝑁)‘⟨𝑅, 𝑆⟩) → 𝑁 ∈ Top)
41	36, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ Top)
42		toptopon2 22420	. . . . . . . 8 ⊢ (𝑁 ∈ Top ↔ 𝑁 ∈ (TopOn‘∪ 𝑁))
43	41, 42	sylib 217	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (TopOn‘∪ 𝑁))
44		cnpf2 22754	. . . . . . 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 1372	. . . . . 6 ⊢ (𝜑 → 𝑂:(𝑋 × 𝑌)⟶∪ 𝑁)
46	45	feqmptd 6961	. . . . 5 ⊢ (𝜑 → 𝑂 = (𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑂‘𝑦)))
47		fveq2 6892	. . . . . 6 ⊢ (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂‘𝑦) = (𝑂‘⟨𝐴, 𝐵⟩))
48		df-ov 7412	. . . . . 6 ⊢ (𝐴𝑂𝐵) = (𝑂‘⟨𝐴, 𝐵⟩)
49	47, 48	eqtr4di 2791	. . . . 5 ⊢ (𝑦 = ⟨𝐴, 𝐵⟩ → (𝑂‘𝑦) = (𝐴𝑂𝐵))
50	9, 39, 46, 49	fmptco 7127	. . . 4 ⊢ (𝜑 → (𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩)) = (𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵)))
51	50	fveq2d 6896	. . 3 ⊢ (𝜑 → ((𝑁 fLimf 𝐿)‘(𝑂 ∘ (𝑥 ∈ 𝑍 ↦ ⟨𝐴, 𝐵⟩))) = ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))
52	38, 51	eleqtrd 2836	. 2 ⊢ (𝜑 → (𝑂‘⟨𝑅, 𝑆⟩) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))
53	1, 52	eqeltrid 2838	1 ⊢ (𝜑 → (𝑅𝑂𝑆) ∈ ((𝑁 fLimf 𝐿)‘(𝑥 ∈ 𝑍 ↦ (𝐴𝑂𝐵))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4635 ∪ cuni 4909 ↦ cmpt 5232 × cxp 5675 ∘ ccom 5681 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 Topctop 22395 TopOnctopon 22412 CnP ccnp 22729 ×_t ctx 23064 Filcfil 23349 fLimf cflf 23439

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-map 8822 df-topgen 17389 df-fbas 20941 df-fg 20942 df-top 22396 df-topon 22413 df-bases 22449 df-ntr 22524 df-nei 22602 df-cnp 22732 df-tx 23066 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444

This theorem is referenced by: tsmsadd 23651

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