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Theorem ismtyval 35958
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 35957 . . 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 5812 . . . . . . . . . 10 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4 xmetf 23482 . . . . . . . . . . 11 (𝑀 ∈ (∞Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ*)
54fdmd 6611 . . . . . . . . . 10 (𝑀 ∈ (∞Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
63, 5sylan9eqr 2800 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑚 = 𝑀) → dom 𝑚 = (𝑋 × 𝑋))
76ad2ant2r 744 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom 𝑚 = (𝑋 × 𝑋))
87dmeqd 5814 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = dom (𝑋 × 𝑋))
9 dmxpid 5839 . . . . . . 7 dom (𝑋 × 𝑋) = 𝑋
108, 9eqtrdi 2794 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = 𝑋)
1110f1oeq2d 6712 . . . . 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))
12 dmeq 5812 . . . . . . . . . 10 (𝑛 = 𝑁 → dom 𝑛 = dom 𝑁)
13 xmetf 23482 . . . . . . . . . . 11 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
1413fdmd 6611 . . . . . . . . . 10 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1512, 14sylan9eqr 2800 . . . . . . . . 9 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑛 = 𝑁) → dom 𝑛 = (𝑌 × 𝑌))
1615ad2ant2l 743 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom 𝑛 = (𝑌 × 𝑌))
1716dmeqd 5814 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = dom (𝑌 × 𝑌))
18 dmxpid 5839 . . . . . . 7 dom (𝑌 × 𝑌) = 𝑌
1917, 18eqtrdi 2794 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = 𝑌)
20 f1oeq3 6706 . . . . . 6 (dom dom 𝑛 = 𝑌 → (𝑓:𝑋1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto𝑌))
2119, 20syl 17 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑓:𝑋1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto𝑌))
2211, 21bitrd 278 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto𝑌))
23 oveq 7281 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
24 oveq 7281 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑓𝑥)𝑛(𝑓𝑦)) = ((𝑓𝑥)𝑁(𝑓𝑦)))
2523, 24eqeqan12d 2752 . . . . . . 7 ((𝑚 = 𝑀𝑛 = 𝑁) → ((𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
2625adantl 482 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → ((𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
2710, 26raleqbidv 3336 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (∀𝑦 ∈ dom dom 𝑚(𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ ∀𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
2810, 27raleqbidv 3336 . . . 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))))
2922, 28anbi12d 631 . . 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)))))
3029abbidv 2807 . 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)))})
31 fvssunirn 6803 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
32 simpl 483 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
3331, 32sselid 3919 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ran ∞Met)
34 fvssunirn 6803 . . 3 (∞Met‘𝑌) ⊆ ran ∞Met
35 simpr 485 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
3634, 35sselid 3919 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ran ∞Met)
37 f1of 6716 . . . . . 6 (𝑓:𝑋1-1-onto𝑌𝑓:𝑋𝑌)
3837adantr 481 . . . . 5 ((𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))) → 𝑓:𝑋𝑌)
39 elfvdm 6806 . . . . . 6 (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
40 elfvdm 6806 . . . . . 6 (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
41 elmapg 8628 . . . . . 6 ((𝑌 ∈ dom ∞Met ∧ 𝑋 ∈ dom ∞Met) → (𝑓 ∈ (𝑌m 𝑋) ↔ 𝑓:𝑋𝑌))
4239, 40, 41syl2anr 597 . . . . 5 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑓 ∈ (𝑌m 𝑋) ↔ 𝑓:𝑋𝑌))
4338, 42syl5ibr 245 . . . 4 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))) → 𝑓 ∈ (𝑌m 𝑋)))
4443abssdv 4002 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ⊆ (𝑌m 𝑋))
45 ovex 7308 . . . 4 (𝑌m 𝑋) ∈ V
4645ssex 5245 . . 3 ({𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ⊆ (𝑌m 𝑋) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ∈ V)
4744, 46syl 17 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ∈ V)
482, 30, 33, 36, 47ovmpod 7425 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 205  wa 396   = wceq 1539  wcel 2106  {cab 2715  wral 3064  Vcvv 3432  wss 3887   cuni 4839   × cxp 5587  dom cdm 5589  ran crn 5590  wf 6429  1-1-ontowf1o 6432  cfv 6433  (class class class)co 7275  cmpo 7277  m cmap 8615  *cxr 11008  ∞Metcxmet 20582   Ismty cismty 35956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-xr 11013  df-xmet 20590  df-ismty 35957
This theorem is referenced by:  isismty  35959
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