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Theorem ismtyval 33904
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:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))})
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.
StepHypRef Expression
1 df-ismty 33903 . . 3 Ismty = (𝑚 ran ∞Met, 𝑛 ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)))})
21a1i 11 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → Ismty = (𝑚 ran ∞Met, 𝑛 ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)))}))
3 dmeq 5471 . . . . . . . . . 10 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4 xmetf 22327 . . . . . . . . . . 11 (𝑀 ∈ (∞Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ*)
5 fdm 6204 . . . . . . . . . . 11 (𝑀:(𝑋 × 𝑋)⟶ℝ* → dom 𝑀 = (𝑋 × 𝑋))
64, 5syl 17 . . . . . . . . . 10 (𝑀 ∈ (∞Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
73, 6sylan9eqr 2808 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑚 = 𝑀) → dom 𝑚 = (𝑋 × 𝑋))
87ad2ant2r 800 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom 𝑚 = (𝑋 × 𝑋))
98dmeqd 5473 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = dom (𝑋 × 𝑋))
10 dmxpid 5492 . . . . . . 7 dom (𝑋 × 𝑋) = 𝑋
119, 10syl6eq 2802 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = 𝑋)
12 f1oeq2 6281 . . . . . 6 (dom dom 𝑚 = 𝑋 → (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto→dom dom 𝑛))
1311, 12syl 17 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto→dom dom 𝑛))
14 dmeq 5471 . . . . . . . . . 10 (𝑛 = 𝑁 → dom 𝑛 = dom 𝑁)
15 xmetf 22327 . . . . . . . . . . 11 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
16 fdm 6204 . . . . . . . . . . 11 (𝑁:(𝑌 × 𝑌)⟶ℝ* → dom 𝑁 = (𝑌 × 𝑌))
1715, 16syl 17 . . . . . . . . . 10 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1814, 17sylan9eqr 2808 . . . . . . . . 9 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑛 = 𝑁) → dom 𝑛 = (𝑌 × 𝑌))
1918ad2ant2l 799 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom 𝑛 = (𝑌 × 𝑌))
2019dmeqd 5473 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = dom (𝑌 × 𝑌))
21 dmxpid 5492 . . . . . . 7 dom (𝑌 × 𝑌) = 𝑌
2220, 21syl6eq 2802 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = 𝑌)
23 f1oeq3 6282 . . . . . 6 (dom dom 𝑛 = 𝑌 → (𝑓:𝑋1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto𝑌))
2422, 23syl 17 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑓:𝑋1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto𝑌))
2513, 24bitrd 268 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto𝑌))
26 oveq 6811 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
27 oveq 6811 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑓𝑥)𝑛(𝑓𝑦)) = ((𝑓𝑥)𝑁(𝑓𝑦)))
2826, 27eqeqan12d 2768 . . . . . . 7 ((𝑚 = 𝑀𝑛 = 𝑁) → ((𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
2928adantl 473 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → ((𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
3011, 29raleqbidv 3283 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
3111, 30raleqbidv 3283 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (∀𝑥 ∈ dom dom 𝑚𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
3225, 31anbi12d 749 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → ((𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦))) ↔ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))))
3332abbidv 2871 . 2 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → {𝑓 ∣ (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛 ∧ ∀𝑥 ∈ dom dom 𝑚𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)))} = {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))})
34 fvssunirn 6370 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
35 simpl 474 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
3634, 35sseldi 3734 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ran ∞Met)
37 fvssunirn 6370 . . 3 (∞Met‘𝑌) ⊆ ran ∞Met
38 simpr 479 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
3937, 38sseldi 3734 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ran ∞Met)
40 f1of 6290 . . . . . 6 (𝑓:𝑋1-1-onto𝑌𝑓:𝑋𝑌)
4140adantr 472 . . . . 5 ((𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))) → 𝑓:𝑋𝑌)
42 elfvdm 6373 . . . . . 6 (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
43 elfvdm 6373 . . . . . 6 (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
44 elmapg 8028 . . . . . 6 ((𝑌 ∈ dom ∞Met ∧ 𝑋 ∈ dom ∞Met) → (𝑓 ∈ (𝑌𝑚 𝑋) ↔ 𝑓:𝑋𝑌))
4542, 43, 44syl2anr 496 . . . . 5 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑓 ∈ (𝑌𝑚 𝑋) ↔ 𝑓:𝑋𝑌))
4641, 45syl5ibr 236 . . . 4 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))) → 𝑓 ∈ (𝑌𝑚 𝑋)))
4746abssdv 3809 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ⊆ (𝑌𝑚 𝑋))
48 ovex 6833 . . . 4 (𝑌𝑚 𝑋) ∈ V
4948ssex 4946 . . 3 ({𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ⊆ (𝑌𝑚 𝑋) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ∈ V)
5047, 49syl 17 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ∈ V)
512, 33, 36, 39, 50ovmpt2d 6945 1 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑀 Ismty 𝑁) = {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1624  wcel 2131  {cab 2738  wral 3042  Vcvv 3332  wss 3707   cuni 4580   × cxp 5256  dom cdm 5258  ran crn 5259  wf 6037  1-1-ontowf1o 6040  cfv 6041  (class class class)co 6805  cmpt2 6807  𝑚 cmap 8015  *cxr 10257  ∞Metcxmt 19925   Ismty cismty 33902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106  ax-cnex 10176  ax-resscn 10177
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-f1 6046  df-fo 6047  df-f1o 6048  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-map 8017  df-xr 10262  df-xmet 19933  df-ismty 33903
This theorem is referenced by:  isismty  33905
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