Theorem ismtyval 37760

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 37759	. . 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 5928	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4		xmetf 24360	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ*)
5	4	fdmd 6757	. . . . . . . . . 10 ⊢ (𝑀 ∈ (∞Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
6	3, 5	sylan9eqr 2802	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑚 = 𝑀) → dom 𝑚 = (𝑋 × 𝑋))
7	6	ad2ant2r 746	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom 𝑚 = (𝑋 × 𝑋))
8	7	dmeqd 5930	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑚 = dom (𝑋 × 𝑋))
9		dmxpid 5955	. . . . . . 7 ⊢ dom (𝑋 × 𝑋) = 𝑋
10	8, 9	eqtrdi 2796	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑚 = 𝑋)
11	10	f1oeq2d 6858	. . . . 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 5928	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → dom 𝑛 = dom 𝑁)
13		xmetf 24360	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
14	13	fdmd 6757	. . . . . . . . . 10 ⊢ (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
15	12, 14	sylan9eqr 2802	. . . . . . . . 9 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑛 = 𝑁) → dom 𝑛 = (𝑌 × 𝑌))
16	15	ad2ant2l 745	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom 𝑛 = (𝑌 × 𝑌))
17	16	dmeqd 5930	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑛 = dom (𝑌 × 𝑌))
18		dmxpid 5955	. . . . . . 7 ⊢ dom (𝑌 × 𝑌) = 𝑌
19	17, 18	eqtrdi 2796	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀 ∧ 𝑛 = 𝑁)) → dom dom 𝑛 = 𝑌)
20		f1oeq3 6852	. . . . . 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 279	. . . 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 7454	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
24		oveq 7454	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))
25	23, 24	eqeqan12d 2754	. . . . . . 7 ⊢ ((𝑚 = 𝑀 ∧ 𝑛 = 𝑁) → ((𝑥𝑚𝑦) = ((𝑓‘𝑥)𝑛(𝑓‘𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))))
26	25	adantl 481	. . . . . 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 3354	. . . . 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 3354	. . . 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 631	. . 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 2811	. 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 6953	. . 3 ⊢ (∞Met‘𝑋) ⊆ ∪ ran ∞Met
32		simpl 482	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
33	31, 32	sselid 4006	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ ∪ ran ∞Met)
34		fvssunirn 6953	. . 3 ⊢ (∞Met‘𝑌) ⊆ ∪ ran ∞Met
35		simpr 484	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
36	34, 35	sselid 4006	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ ∪ ran ∞Met)
37		f1of 6862	. . . . . 6 ⊢ (𝑓:𝑋–1-1-onto→𝑌 → 𝑓:𝑋⟶𝑌)
38	37	adantr 480	. . . . 5 ⊢ ((𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))) → 𝑓:𝑋⟶𝑌)
39		elfvdm 6957	. . . . . 6 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
40		elfvdm 6957	. . . . . 6 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
41		elmapg 8897	. . . . . 6 ⊢ ((𝑌 ∈ dom ∞Met ∧ 𝑋 ∈ dom ∞Met) → (𝑓 ∈ (𝑌 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝑌))
42	39, 40, 41	syl2anr 596	. . . . 5 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑓 ∈ (𝑌 ↑m 𝑋) ↔ 𝑓:𝑋⟶𝑌))
43	38, 42	imbitrrid 246	. . . 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 4091	. . 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 7481	. . . 4 ⊢ (𝑌 ↑m 𝑋) ∈ V
46	45	ssex 5339	. . 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 7602	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 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  Vcvv 3488   ⊆ wss 3976  ∪ cuni 4931   × cxp 5698  dom cdm 5700  ran crn 5701  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450   ↑_m cmap 8884  ℝ^*cxr 11323  ∞Metcxmet 21372   Ismty cismty 37758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-xr 11328  df-xmet 21380  df-ismty 37759

This theorem is referenced by:  isismty  37761