Theorem ismtyval 35093

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 35092	. . 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 5772	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4		xmetf 22939	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ*)
5	4	fdmd 6523	. . . . . . . . . 10 ⊢ (𝑀 ∈ (∞Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
6	3, 5	sylan9eqr 2878	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑚 = 𝑀) → dom 𝑚 = (𝑋 × 𝑋))
7	6	ad2ant2r 745	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom 𝑚 = (𝑋 × 𝑋))
8	7	dmeqd 5774	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑚 = dom (𝑋 × 𝑋))
9		dmxpid 5800	. . . . . . 7 ⊢ dom (𝑋 × 𝑋) = 𝑋
10	8, 9	syl6eq 2872	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑚 = 𝑋)
11	10	f1oeq2d 6611	. . . . 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 5772	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → dom 𝑛 = dom 𝑁)
13		xmetf 22939	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
14	13	fdmd 6523	. . . . . . . . . 10 ⊢ (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
15	12, 14	sylan9eqr 2878	. . . . . . . . 9 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑛 = 𝑁) → dom 𝑛 = (𝑌 × 𝑌))
16	15	ad2ant2l 744	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom 𝑛 = (𝑌 × 𝑌))
17	16	dmeqd 5774	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑛 = dom (𝑌 × 𝑌))
18		dmxpid 5800	. . . . . . 7 ⊢ dom (𝑌 × 𝑌) = 𝑌
19	17, 18	syl6eq 2872	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑛 = 𝑌)
20		f1oeq3 6606	. . . . . 6 ⊢ (dom dom 𝑛 = 𝑌 → (𝑓:𝑋–1-1-onto→dom dom 𝑛 ↔ 𝑓:𝑋–1-1-onto→𝑌))
21	19, 20	syl 17	. . . . 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 281	. . . 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 7162	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
24		oveq 7162	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))
25	23, 24	eqeqan12d 2838	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → ((𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))))
26	25	adantl 484	. . . . . 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 3401	. . . . 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 3401	. . . 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 632	. . 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 2885	. 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 6699	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
32		simpl 485	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
33	31, 32	sseldi 3965	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ ∪ ran ∞Met)
34		fvssunirn 6699	. . 3 ⊢ (∞Met‘𝑌) ⊆ ∪ ran ∞Met
35		simpr 487	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
36	34, 35	sseldi 3965	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ ∪ ran ∞Met)
37		f1of 6615	. . . . . 6 ⊢ (𝑓:𝑋–1-1-onto→𝑌 → 𝑓:𝑋⟶𝑌)
38	37	adantr 483	. . . . 5 ⊢ ((𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))) → 𝑓:𝑋⟶𝑌)
39		elfvdm 6702	. . . . . 6 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
40		elfvdm 6702	. . . . . 6 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
41		elmapg 8419	. . . . . 6 ⊢ ((𝑌 ∈ dom ∞Met ∧ 𝑋 ∈ dom ∞Met) → (𝑓 ∈ (𝑌 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝑌))
42	39, 40, 41	syl2anr 598	. . . . 5 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑓 ∈ (𝑌 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝑌))
43	38, 42	syl5ibr 248	. . . 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 4045	. . 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 7189	. . . 4 ⊢ (𝑌 ↑m 𝑋) ∈ V
46	45	ssex 5225	. . 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 17	. 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 7302	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 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  {cab 2799  ∀wral 3138  Vcvv 3494   ⊆ wss 3936  ∪ cuni 4838   × cxp 5553  dom cdm 5555  ran crn 5556  ⟶wf 6351  –1-1-onto→wf1o 6354  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158   ↑_m cmap 8406  ℝ^*cxr 10674  ∞Metcxmet 20530   Ismty cismty 35091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8408  df-xr 10679  df-xmet 20538  df-ismty 35092

This theorem is referenced by:  isismty  35094