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Theorem ismtyval 35150
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 35149 . . 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 5760 . . . . . . . . . 10 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4 xmetf 22934 . . . . . . . . . . 11 (𝑀 ∈ (∞Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ*)
54fdmd 6512 . . . . . . . . . 10 (𝑀 ∈ (∞Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
63, 5sylan9eqr 2881 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑚 = 𝑀) → dom 𝑚 = (𝑋 × 𝑋))
76ad2ant2r 746 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom 𝑚 = (𝑋 × 𝑋))
87dmeqd 5762 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = dom (𝑋 × 𝑋))
9 dmxpid 5788 . . . . . . 7 dom (𝑋 × 𝑋) = 𝑋
108, 9syl6eq 2875 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = 𝑋)
1110f1oeq2d 6600 . . . . 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 5760 . . . . . . . . . 10 (𝑛 = 𝑁 → dom 𝑛 = dom 𝑁)
13 xmetf 22934 . . . . . . . . . . 11 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
1413fdmd 6512 . . . . . . . . . 10 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1512, 14sylan9eqr 2881 . . . . . . . . 9 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑛 = 𝑁) → dom 𝑛 = (𝑌 × 𝑌))
1615ad2ant2l 745 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom 𝑛 = (𝑌 × 𝑌))
1716dmeqd 5762 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = dom (𝑌 × 𝑌))
18 dmxpid 5788 . . . . . . 7 dom (𝑌 × 𝑌) = 𝑌
1917, 18syl6eq 2875 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = 𝑌)
20 f1oeq3 6595 . . . . . 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 282 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → (𝑓:dom dom 𝑚1-1-onto→dom dom 𝑛𝑓:𝑋1-1-onto𝑌))
23 oveq 7152 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
24 oveq 7152 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑓𝑥)𝑛(𝑓𝑦)) = ((𝑓𝑥)𝑁(𝑓𝑦)))
2523, 24eqeqan12d 2841 . . . . . . 7 ((𝑚 = 𝑀𝑛 = 𝑁) → ((𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
2625adantl 485 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → ((𝑥𝑚𝑦) = ((𝑓𝑥)𝑛(𝑓𝑦)) ↔ (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))))
2710, 26raleqbidv 3393 . . . . 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 3393 . . . 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 633 . . 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 2888 . 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 6688 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
32 simpl 486 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
3331, 32sseldi 3951 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ran ∞Met)
34 fvssunirn 6688 . . 3 (∞Met‘𝑌) ⊆ ran ∞Met
35 simpr 488 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
3634, 35sseldi 3951 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ran ∞Met)
37 f1of 6604 . . . . . 6 (𝑓:𝑋1-1-onto𝑌𝑓:𝑋𝑌)
3837adantr 484 . . . . 5 ((𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))) → 𝑓:𝑋𝑌)
39 elfvdm 6691 . . . . . 6 (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
40 elfvdm 6691 . . . . . 6 (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
41 elmapg 8411 . . . . . 6 ((𝑌 ∈ dom ∞Met ∧ 𝑋 ∈ dom ∞Met) → (𝑓 ∈ (𝑌m 𝑋) ↔ 𝑓:𝑋𝑌))
4239, 40, 41syl2anr 599 . . . . 5 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑓 ∈ (𝑌m 𝑋) ↔ 𝑓:𝑋𝑌))
4338, 42syl5ibr 249 . . . 4 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))) → 𝑓 ∈ (𝑌m 𝑋)))
4443abssdv 4031 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ⊆ (𝑌m 𝑋))
45 ovex 7179 . . . 4 (𝑌m 𝑋) ∈ V
4645ssex 5212 . . 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 7292 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 209  wa 399   = wceq 1538  wcel 2115  {cab 2802  wral 3133  Vcvv 3480  wss 3919   cuni 4825   × cxp 5541  dom cdm 5543  ran crn 5544  wf 6340  1-1-ontowf1o 6343  cfv 6344  (class class class)co 7146  cmpo 7148  m cmap 8398  *cxr 10668  ∞Metcxmet 20525   Ismty cismty 35148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-cnex 10587  ax-resscn 10588
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-map 8400  df-xr 10673  df-xmet 20533  df-ismty 35149
This theorem is referenced by:  isismty  35151
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