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Theorem ismtyval 33912
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 33911 . . 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 5479 . . . . . . . . . 10 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4 xmetf 22335 . . . . . . . . . . 11 (𝑀 ∈ (∞Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ*)
5 fdm 6212 . . . . . . . . . . 11 (𝑀:(𝑋 × 𝑋)⟶ℝ* → dom 𝑀 = (𝑋 × 𝑋))
64, 5syl 17 . . . . . . . . . 10 (𝑀 ∈ (∞Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
73, 6sylan9eqr 2816 . . . . . . . . 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 5481 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = dom (𝑋 × 𝑋))
10 dmxpid 5500 . . . . . . 7 dom (𝑋 × 𝑋) = 𝑋
119, 10syl6eq 2810 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = 𝑋)
12 f1oeq2 6289 . . . . . 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 5479 . . . . . . . . . 10 (𝑛 = 𝑁 → dom 𝑛 = dom 𝑁)
15 xmetf 22335 . . . . . . . . . . 11 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
16 fdm 6212 . . . . . . . . . . 11 (𝑁:(𝑌 × 𝑌)⟶ℝ* → dom 𝑁 = (𝑌 × 𝑌))
1715, 16syl 17 . . . . . . . . . 10 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1814, 17sylan9eqr 2816 . . . . . . . . 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 5481 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = dom (𝑌 × 𝑌))
21 dmxpid 5500 . . . . . . 7 dom (𝑌 × 𝑌) = 𝑌
2220, 21syl6eq 2810 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = 𝑌)
23 f1oeq3 6290 . . . . . 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 6819 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
27 oveq 6819 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑓𝑥)𝑛(𝑓𝑦)) = ((𝑓𝑥)𝑁(𝑓𝑦)))
2826, 27eqeqan12d 2776 . . . . . . 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 3291 . . . . 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 3291 . . . 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 2879 . 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 6378 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
35 simpl 474 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
3634, 35sseldi 3742 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ran ∞Met)
37 fvssunirn 6378 . . 3 (∞Met‘𝑌) ⊆ ran ∞Met
38 simpr 479 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
3937, 38sseldi 3742 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ran ∞Met)
40 f1of 6298 . . . . . 6 (𝑓:𝑋1-1-onto𝑌𝑓:𝑋𝑌)
4140adantr 472 . . . . 5 ((𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))) → 𝑓:𝑋𝑌)
42 elfvdm 6381 . . . . . 6 (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
43 elfvdm 6381 . . . . . 6 (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
44 elmapg 8036 . . . . . 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 3817 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ⊆ (𝑌𝑚 𝑋))
48 ovex 6841 . . . 4 (𝑌𝑚 𝑋) ∈ V
4948ssex 4954 . . 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 6953 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 1632  wcel 2139  {cab 2746  wral 3050  Vcvv 3340  wss 3715   cuni 4588   × cxp 5264  dom cdm 5266  ran crn 5267  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  cmpt2 6815  𝑚 cmap 8023  *cxr 10265  ∞Metcxmt 19933   Ismty cismty 33910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-map 8025  df-xr 10270  df-xmet 19941  df-ismty 33911
This theorem is referenced by:  isismty  33913
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