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Theorem ismtyval 35604
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 35603 . . 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 5747 . . . . . . . . . 10 (𝑚 = 𝑀 → dom 𝑚 = dom 𝑀)
4 xmetf 23085 . . . . . . . . . . 11 (𝑀 ∈ (∞Met‘𝑋) → 𝑀:(𝑋 × 𝑋)⟶ℝ*)
54fdmd 6516 . . . . . . . . . 10 (𝑀 ∈ (∞Met‘𝑋) → dom 𝑀 = (𝑋 × 𝑋))
63, 5sylan9eqr 2796 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑚 = 𝑀) → dom 𝑚 = (𝑋 × 𝑋))
76ad2ant2r 747 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom 𝑚 = (𝑋 × 𝑋))
87dmeqd 5749 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = dom (𝑋 × 𝑋))
9 dmxpid 5774 . . . . . . 7 dom (𝑋 × 𝑋) = 𝑋
108, 9eqtrdi 2790 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑚 = 𝑋)
1110f1oeq2d 6617 . . . . 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 5747 . . . . . . . . . 10 (𝑛 = 𝑁 → dom 𝑛 = dom 𝑁)
13 xmetf 23085 . . . . . . . . . . 11 (𝑁 ∈ (∞Met‘𝑌) → 𝑁:(𝑌 × 𝑌)⟶ℝ*)
1413fdmd 6516 . . . . . . . . . 10 (𝑁 ∈ (∞Met‘𝑌) → dom 𝑁 = (𝑌 × 𝑌))
1512, 14sylan9eqr 2796 . . . . . . . . 9 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑛 = 𝑁) → dom 𝑛 = (𝑌 × 𝑌))
1615ad2ant2l 746 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom 𝑛 = (𝑌 × 𝑌))
1716dmeqd 5749 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = dom (𝑌 × 𝑌))
18 dmxpid 5774 . . . . . . 7 dom (𝑌 × 𝑌) = 𝑌
1917, 18eqtrdi 2790 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) ∧ (𝑚 = 𝑀𝑛 = 𝑁)) → dom dom 𝑛 = 𝑌)
20 f1oeq3 6611 . . . . . 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 7179 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥𝑚𝑦) = (𝑥𝑀𝑦))
24 oveq 7179 . . . . . . . 8 (𝑛 = 𝑁 → ((𝑓𝑥)𝑛(𝑓𝑦)) = ((𝑓𝑥)𝑁(𝑓𝑦)))
2523, 24eqeqan12d 2756 . . . . . . 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 3305 . . . . 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 3305 . . . 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 634 . . 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 2803 . 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 6706 . . 3 (∞Met‘𝑋) ⊆ ran ∞Met
32 simpl 486 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ∈ (∞Met‘𝑋))
3331, 32sseldi 3876 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑀 ran ∞Met)
34 fvssunirn 6706 . . 3 (∞Met‘𝑌) ⊆ ran ∞Met
35 simpr 488 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ∈ (∞Met‘𝑌))
3634, 35sseldi 3876 . 2 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝑁 ran ∞Met)
37 f1of 6621 . . . . . 6 (𝑓:𝑋1-1-onto𝑌𝑓:𝑋𝑌)
3837adantr 484 . . . . 5 ((𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦))) → 𝑓:𝑋𝑌)
39 elfvdm 6709 . . . . . 6 (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
40 elfvdm 6709 . . . . . 6 (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
41 elmapg 8453 . . . . . 6 ((𝑌 ∈ dom ∞Met ∧ 𝑋 ∈ dom ∞Met) → (𝑓 ∈ (𝑌m 𝑋) ↔ 𝑓:𝑋𝑌))
4239, 40, 41syl2anr 600 . . . . 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 3959 . . 3 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → {𝑓 ∣ (𝑓:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝑓𝑥)𝑁(𝑓𝑦)))} ⊆ (𝑌m 𝑋))
45 ovex 7206 . . . 4 (𝑌m 𝑋) ∈ V
4645ssex 5190 . . 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 7320 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 1542  wcel 2114  {cab 2717  wral 3054  Vcvv 3399  wss 3844   cuni 4797   × cxp 5524  dom cdm 5526  ran crn 5527  wf 6336  1-1-ontowf1o 6339  cfv 6340  (class class class)co 7173  cmpo 7175  m cmap 8440  *cxr 10755  ∞Metcxmet 20205   Ismty cismty 35602
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482  ax-cnex 10674  ax-resscn 10675
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7176  df-oprab 7177  df-mpo 7178  df-map 8442  df-xr 10760  df-xmet 20213  df-ismty 35603
This theorem is referenced by:  isismty  35605
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