ismtybndlem - Mathbox for Jeff Madsen

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Theorem ismtybndlem 36662

Description: Lemma for ismtybnd 36663. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 19-Jan-2014.)

**Assertion**
Ref	Expression
ismtybndlem	⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝑀 ∈ (Bnd‘𝑋) → 𝑁 ∈ (Bnd‘𝑌)))

Proof of Theorem ismtybndlem Dummy variables 𝑤 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isismty 36657	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧𝑀𝑤) = ((𝐹‘𝑧)𝑁(𝐹‘𝑤)))))
2	1	biimp3a 1469	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧𝑀𝑤) = ((𝐹‘𝑧)𝑁(𝐹‘𝑤))))
3	2	simpld 495	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → 𝐹:𝑋–1-1-onto→𝑌)
4		f1ocnv 6842	. . . . . . . . . . 11 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
5		f1of 6830	. . . . . . . . . . 11 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
6	3, 4, 5	3syl 18	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → ^◡𝐹:𝑌⟶𝑋)
7	6	ffvelcdmda 7083	. . . . . . . . 9 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ 𝑦 ∈ 𝑌) → (^◡𝐹‘𝑦) ∈ 𝑋)
8		oveq1 7412	. . . . . . . . . . . 12 ⊢ (𝑥 = (^◡𝐹‘𝑦) → (𝑥(ball‘𝑀)𝑟) = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟))
9	8	eqeq2d 2743	. . . . . . . . . . 11 ⊢ (𝑥 = (^◡𝐹‘𝑦) → (𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ 𝑋 = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)))
10	9	rexbidv 3178	. . . . . . . . . 10 ⊢ (𝑥 = (^◡𝐹‘𝑦) → (∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) ↔ ∃𝑟 ∈ ℝ⁺ 𝑋 = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)))
11	10	rspcv 3608	. . . . . . . . 9 ⊢ ((^◡𝐹‘𝑦) ∈ 𝑋 → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ⁺ 𝑋 = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)))
12	7, 11	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ 𝑦 ∈ 𝑌) → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ⁺ 𝑋 = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)))
13		imaeq2 6053	. . . . . . . . . . 11 ⊢ (𝑋 = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟) → (𝐹 “ 𝑋) = (𝐹 “ ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)))
14		f1ofo 6837	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌)
15		foima 6807	. . . . . . . . . . . . . 14 ⊢ (𝐹:𝑋–onto→𝑌 → (𝐹 “ 𝑋) = 𝑌)
16	3, 14, 15	3syl 18	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝐹 “ 𝑋) = 𝑌)
17	16	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝐹 “ 𝑋) = 𝑌)
18		rpxr 12979	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 ∈ ℝ⁺ → 𝑟 ∈ ℝ^*)
19	18	adantl 482	. . . . . . . . . . . . . . 15 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ 𝑟 ∈ ℝ⁺) → 𝑟 ∈ ℝ^*)
20	7, 19	anim12dan 619	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((^◡𝐹‘𝑦) ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*))
21		ismtyima 36659	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ ((^◡𝐹‘𝑦) ∈ 𝑋 ∧ 𝑟 ∈ ℝ^*)) → (𝐹 “ ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)) = ((𝐹‘(^◡𝐹‘𝑦))(ball‘𝑁)𝑟))
22	20, 21	syldan 591	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝐹 “ ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)) = ((𝐹‘(^◡𝐹‘𝑦))(ball‘𝑁)𝑟))
23		simpl 483	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺) → 𝑦 ∈ 𝑌)
24		f1ocnvfv2 7271	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑦 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑦)) = 𝑦)
25	3, 23, 24	syl2an 596	. . . . . . . . . . . . . 14 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝐹‘(^◡𝐹‘𝑦)) = 𝑦)
26	25	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((𝐹‘(^◡𝐹‘𝑦))(ball‘𝑁)𝑟) = (𝑦(ball‘𝑁)𝑟))
27	22, 26	eqtrd 2772	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝐹 “ ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)) = (𝑦(ball‘𝑁)𝑟))
28	17, 27	eqeq12d 2748	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → ((𝐹 “ 𝑋) = (𝐹 “ ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟)) ↔ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
29	13, 28	imbitrid 243	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑦 ∈ 𝑌 ∧ 𝑟 ∈ ℝ⁺)) → (𝑋 = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟) → 𝑌 = (𝑦(ball‘𝑁)𝑟)))
30	29	anassrs 468	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ 𝑦 ∈ 𝑌) ∧ 𝑟 ∈ ℝ⁺) → (𝑋 = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟) → 𝑌 = (𝑦(ball‘𝑁)𝑟)))
31	30	reximdva 3168	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ 𝑦 ∈ 𝑌) → (∃𝑟 ∈ ℝ⁺ 𝑋 = ((^◡𝐹‘𝑦)(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
32	12, 31	syld 47	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ 𝑦 ∈ 𝑌) → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
33	32	ralrimdva 3154	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → ∀𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
34		simp2 1137	. . . . . 6 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → 𝑁 ∈ (∞Met‘𝑌))
35	33, 34	jctild 526	. . . . 5 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑁 ∈ (∞Met‘𝑌) ∧ ∀𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟))))
36	35	3expib 1122	. . . 4 ⊢ (𝑀 ∈ (∞Met‘𝑋) → ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑁 ∈ (∞Met‘𝑌) ∧ ∀𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))))
37	36	com12 32	. . 3 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝑀 ∈ (∞Met‘𝑋) → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟) → (𝑁 ∈ (∞Met‘𝑌) ∧ ∀𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))))
38	37	impd 411	. 2 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → ((𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → (𝑁 ∈ (∞Met‘𝑌) ∧ ∀𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟))))
39		isbndx 36638	. 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (∞Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ⁺ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
40		isbndx 36638	. 2 ⊢ (𝑁 ∈ (Bnd‘𝑌) ↔ (𝑁 ∈ (∞Met‘𝑌) ∧ ∀𝑦 ∈ 𝑌 ∃𝑟 ∈ ℝ⁺ 𝑌 = (𝑦(ball‘𝑁)𝑟)))
41	38, 39, 40	3imtr4g 295	1 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝑀 ∈ (Bnd‘𝑋) → 𝑁 ∈ (Bnd‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 ^◡ccnv 5674 “ cima 5678 ⟶wf 6536 –onto→wfo 6538 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7405 ℝ^*cxr 11243 ℝ⁺crp 12970 ∞Metcxmet 20921 ballcbl 20923 Bndcbnd 36623 Ismty cismty 36654

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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-po 5587 df-so 5588 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7971 df-2nd 7972 df-er 8699 df-ec 8701 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-2 12271 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-bnd 36635 df-ismty 36655

This theorem is referenced by: ismtybnd 36663

W3C validator