Theorem ismtyval 38432

Description: The set of isometries between two metric spaces. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
ismtyval	⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑀 Ismty 𝑁) = {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))})

Distinct variable groups:   𝑓,𝑀,𝑥,𝑦   𝑓,𝑁,𝑥,𝑦   𝑓,𝑋,𝑥,𝑦   𝑓,𝑌,𝑥,𝑦

Proof of Theorem ismtyval

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ismty 38431	. . 3 ⊢ Ismty = (𝑚 ∈ ∪ ran ∞Met, 𝑛 ∈ ∪ ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom 𝑚–1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)))})
2	1	a1i 11	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → Ismty = (𝑚 ∈ ∪ ran ∞Met, 𝑛 ∈ ∪ ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom 𝑚–1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)))}))
3		dmeq 5895	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4		xmetf 24467	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ*)
5	4	fdmd 6718	. . . . . . . . . 10 ⊢ (𝑀 ∈ (∞Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
6	3, 5	sylan9eqr 2820	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑚 = 𝑀) → dom 𝑚 = (𝑋 × 𝑋))
7	6	ad2ant2r 759	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom 𝑚 = (𝑋 × 𝑋))
8	7	dmeqd 5897	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑚 = dom (𝑋 × 𝑋))
9		dmxpid 5922	. . . . . . 7 ⊢ dom (𝑋 × 𝑋) = 𝑋
10	8, 9	eqtrdi 2814	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑚 = 𝑋)
11	10	f1oeq2d 6818	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (𝑓:dom dom 𝑚–1-1-onto→dom dom 𝑛 ↔ 𝑓:𝑋–1-1-onto→dom dom 𝑛))
12		dmeq 5895	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → dom 𝑛 = dom 𝑁)
13		xmetf 24467	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
14	13	fdmd 6718	. . . . . . . . . 10 ⊢ (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
15	12, 14	sylan9eqr 2820	. . . . . . . . 9 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑛 = 𝑁) → dom 𝑛 = (𝑌 × 𝑌))
16	15	ad2ant2l 758	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom 𝑛 = (𝑌 × 𝑌))
17	16	dmeqd 5897	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑛 = dom (𝑌 × 𝑌))
18		dmxpid 5922	. . . . . . 7 ⊢ dom (𝑌 × 𝑌) = 𝑌
19	17, 18	eqtrdi 2814	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑛 = 𝑌)
20		f1oeq3 6812	. . . . . 6 ⊢ (dom dom 𝑛 = 𝑌 → (𝑓:𝑋–1-1-onto→dom dom 𝑛 ↔ 𝑓:𝑋–1-1-onto→𝑌))
21	19, 20	syl 18	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (𝑓:𝑋–1-1-onto→dom dom 𝑛 ↔ 𝑓:𝑋–1-1-onto→𝑌))
22	11, 21	bitrd 282	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (𝑓:dom dom 𝑚–1-1-onto→dom dom 𝑛 ↔ 𝑓:𝑋–1-1-onto→𝑌))
23		oveq 7418	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
24		oveq 7418	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))
25	23, 24	eqeqan12d 2777	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → ((𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))))
26	25	adantl 486	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → ((𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))))
27	10, 26	raleqbidv 3338	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))))
28	10, 27	raleqbidv 3338	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → (∀𝑥 ∈ dom dom 𝑚∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))))
29	22, 28	anbi12d 643	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → ((𝑓:dom dom 𝑚–1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦))) ↔ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))))
30	29	abbidv 2829	. 2 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → {𝑓 ∣ (𝑓:dom dom 𝑚–1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)))} = {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))})
31		fvssunirn 6914	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
32		simpl 487	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
33	31, 32	sselid 3936	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ ∪ ran ∞Met)
34		fvssunirn 6914	. . 3 ⊢ (∞Met‘𝑌) ⊆ ∪ ran ∞Met
35		simpr 489	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
36	34, 35	sselid 3936	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ ∪ ran ∞Met)
37		f1of 6822	. . . . . 6 ⊢ (𝑓:𝑋–1-1-onto→𝑌 → 𝑓:𝑋⟶𝑌)
38	37	adantr 485	. . . . 5 ⊢ ((𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))) → 𝑓:𝑋⟶𝑌)
39		elfvdm 6917	. . . . . 6 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
40		elfvdm 6917	. . . . . 6 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
41		elmapg 8837	. . . . . 6 ⊢ ((𝑌 ∈ dom ∞Met ∧ 𝑋 ∈ dom ∞Met) → (𝑓 ∈ (𝑌 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝑌))
42	39, 40, 41	syl2anr 608	. . . . 5 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑓 ∈ (𝑌 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝑌))
43	38, 42	imbitrrid 249	. . . 4 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))) → 𝑓 ∈ (𝑌 ↑m 𝑋)))
44	43	abssdv 4022	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))} ⊆ (𝑌 ↑m 𝑋))
45		ovex 7445	. . . 4 ⊢ (𝑌 ↑m 𝑋) ∈ V
46	45	ssex 5292	. . 3 ⊢ ({𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))} ⊆ (𝑌 ↑m 𝑋) → {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))} ∈ V)
47	44, 46	syl 18	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))} ∈ V)
48	2, 30, 33, 36, 47	ovmpod 7564	1 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑀 Ismty 𝑁) = {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {cab 2741  ∀wral 3079  Vcvv 3455   ⊆ wss 3906  ∪ cuni 4873   × cxp 5661  dom cdm 5663  ran crn 5664  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414   ↑_m cmap 8825  ℝ^*cxr 11243  ∞Metcxmet 21488   Ismty cismty 38430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-map 8827  df-xr 11248  df-xmet 21496  df-ismty 38431

This theorem is referenced by:  isismty  38433