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Theorem ismtyval 36656
Description: The set of isometries between two metric spaces. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
ismtyval ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ (𝑀 Ismty 𝑁) = {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))})
Distinct variable groups:   𝑓,𝑀,π‘₯,𝑦   𝑓,𝑁,π‘₯,𝑦   𝑓,𝑋,π‘₯,𝑦   𝑓,π‘Œ,π‘₯,𝑦
Proof of Theorem ismtyval
Dummy variables π‘š 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ismty 36655 . . 3 Ismty = (π‘š ∈ βˆͺ ran ∞Met, 𝑛 ∈ βˆͺ ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom π‘šβ€“1-1-ontoβ†’dom dom 𝑛 ∧ βˆ€π‘₯ ∈ dom dom π‘šβˆ€π‘¦ ∈ dom dom π‘š(π‘₯π‘šπ‘¦) = ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦)))})
21a1i 11 . 2 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ Ismty = (π‘š ∈ βˆͺ ran ∞Met, 𝑛 ∈ βˆͺ ran ∞Met ↦ {𝑓 ∣ (𝑓:dom dom π‘šβ€“1-1-ontoβ†’dom dom 𝑛 ∧ βˆ€π‘₯ ∈ dom dom π‘šβˆ€π‘¦ ∈ dom dom π‘š(π‘₯π‘šπ‘¦) = ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦)))}))
3 dmeq 5901 . . . . . . . . . 10 (π‘š = 𝑀 β†’ dom π‘š = dom 𝑀)
4 xmetf 23826 . . . . . . . . . . 11 (𝑀 ∈ (∞Metβ€˜π‘‹) β†’ 𝑀:(𝑋 Γ— 𝑋)βŸΆβ„*)
54fdmd 6725 . . . . . . . . . 10 (𝑀 ∈ (∞Metβ€˜π‘‹) β†’ dom 𝑀 = (𝑋 Γ— 𝑋))
63, 5sylan9eqr 2794 . . . . . . . . 9 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ π‘š = 𝑀) β†’ dom π‘š = (𝑋 Γ— 𝑋))
76ad2ant2r 745 . . . . . . . 8 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ dom π‘š = (𝑋 Γ— 𝑋))
87dmeqd 5903 . . . . . . 7 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ dom dom π‘š = dom (𝑋 Γ— 𝑋))
9 dmxpid 5927 . . . . . . 7 dom (𝑋 Γ— 𝑋) = 𝑋
108, 9eqtrdi 2788 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ dom dom π‘š = 𝑋)
1110f1oeq2d 6826 . . . . 5 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ (𝑓:dom dom π‘šβ€“1-1-ontoβ†’dom dom 𝑛 ↔ 𝑓:𝑋–1-1-ontoβ†’dom dom 𝑛))
12 dmeq 5901 . . . . . . . . . 10 (𝑛 = 𝑁 β†’ dom 𝑛 = dom 𝑁)
13 xmetf 23826 . . . . . . . . . . 11 (𝑁 ∈ (∞Metβ€˜π‘Œ) β†’ 𝑁:(π‘Œ Γ— π‘Œ)βŸΆβ„*)
1413fdmd 6725 . . . . . . . . . 10 (𝑁 ∈ (∞Metβ€˜π‘Œ) β†’ dom 𝑁 = (π‘Œ Γ— π‘Œ))
1512, 14sylan9eqr 2794 . . . . . . . . 9 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ 𝑛 = 𝑁) β†’ dom 𝑛 = (π‘Œ Γ— π‘Œ))
1615ad2ant2l 744 . . . . . . . 8 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ dom 𝑛 = (π‘Œ Γ— π‘Œ))
1716dmeqd 5903 . . . . . . 7 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ dom dom 𝑛 = dom (π‘Œ Γ— π‘Œ))
18 dmxpid 5927 . . . . . . 7 dom (π‘Œ Γ— π‘Œ) = π‘Œ
1917, 18eqtrdi 2788 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ dom dom 𝑛 = π‘Œ)
20 f1oeq3 6820 . . . . . 6 (dom dom 𝑛 = π‘Œ β†’ (𝑓:𝑋–1-1-ontoβ†’dom dom 𝑛 ↔ 𝑓:𝑋–1-1-ontoβ†’π‘Œ))
2119, 20syl 17 . . . . 5 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ (𝑓:𝑋–1-1-ontoβ†’dom dom 𝑛 ↔ 𝑓:𝑋–1-1-ontoβ†’π‘Œ))
2211, 21bitrd 278 . . . 4 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ (𝑓:dom dom π‘šβ€“1-1-ontoβ†’dom dom 𝑛 ↔ 𝑓:𝑋–1-1-ontoβ†’π‘Œ))
23 oveq 7411 . . . . . . . 8 (π‘š = 𝑀 β†’ (π‘₯π‘šπ‘¦) = (π‘₯𝑀𝑦))
24 oveq 7411 . . . . . . . 8 (𝑛 = 𝑁 β†’ ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦)) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))
2523, 24eqeqan12d 2746 . . . . . . 7 ((π‘š = 𝑀 ∧ 𝑛 = 𝑁) β†’ ((π‘₯π‘šπ‘¦) = ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦)) ↔ (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦))))
2625adantl 482 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ ((π‘₯π‘šπ‘¦) = ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦)) ↔ (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦))))
2710, 26raleqbidv 3342 . . . . 5 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ (βˆ€π‘¦ ∈ dom dom π‘š(π‘₯π‘šπ‘¦) = ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦)) ↔ βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦))))
2810, 27raleqbidv 3342 . . . 4 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ (βˆ€π‘₯ ∈ dom dom π‘šβˆ€π‘¦ ∈ dom dom π‘š(π‘₯π‘šπ‘¦) = ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦))))
2922, 28anbi12d 631 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ ((𝑓:dom dom π‘šβ€“1-1-ontoβ†’dom dom 𝑛 ∧ βˆ€π‘₯ ∈ dom dom π‘šβˆ€π‘¦ ∈ dom dom π‘š(π‘₯π‘šπ‘¦) = ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦))) ↔ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))))
3029abbidv 2801 . 2 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (π‘š = 𝑀 ∧ 𝑛 = 𝑁)) β†’ {𝑓 ∣ (𝑓:dom dom π‘šβ€“1-1-ontoβ†’dom dom 𝑛 ∧ βˆ€π‘₯ ∈ dom dom π‘šβˆ€π‘¦ ∈ dom dom π‘š(π‘₯π‘šπ‘¦) = ((π‘“β€˜π‘₯)𝑛(π‘“β€˜π‘¦)))} = {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))})
31 fvssunirn 6921 . . 3 (∞Metβ€˜π‘‹) βŠ† βˆͺ ran ∞Met
32 simpl 483 . . 3 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
3331, 32sselid 3979 . 2 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ 𝑀 ∈ βˆͺ ran ∞Met)
34 fvssunirn 6921 . . 3 (∞Metβ€˜π‘Œ) βŠ† βˆͺ ran ∞Met
35 simpr 485 . . 3 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))
3634, 35sselid 3979 . 2 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ 𝑁 ∈ βˆͺ ran ∞Met)
37 f1of 6830 . . . . . 6 (𝑓:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝑓:π‘‹βŸΆπ‘Œ)
3837adantr 481 . . . . 5 ((𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦))) β†’ 𝑓:π‘‹βŸΆπ‘Œ)
39 elfvdm 6925 . . . . . 6 (𝑁 ∈ (∞Metβ€˜π‘Œ) β†’ π‘Œ ∈ dom ∞Met)
40 elfvdm 6925 . . . . . 6 (𝑀 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 ∈ dom ∞Met)
41 elmapg 8829 . . . . . 6 ((π‘Œ ∈ dom ∞Met ∧ 𝑋 ∈ dom ∞Met) β†’ (𝑓 ∈ (π‘Œ ↑m 𝑋) ↔ 𝑓:π‘‹βŸΆπ‘Œ))
4239, 40, 41syl2anr 597 . . . . 5 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ (𝑓 ∈ (π‘Œ ↑m 𝑋) ↔ 𝑓:π‘‹βŸΆπ‘Œ))
4338, 42imbitrrid 245 . . . 4 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ ((𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦))) β†’ 𝑓 ∈ (π‘Œ ↑m 𝑋)))
4443abssdv 4064 . . 3 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))} βŠ† (π‘Œ ↑m 𝑋))
45 ovex 7438 . . . 4 (π‘Œ ↑m 𝑋) ∈ V
4645ssex 5320 . . 3 ({𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))} βŠ† (π‘Œ ↑m 𝑋) β†’ {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))} ∈ V)
4744, 46syl 17 . 2 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))} ∈ V)
482, 30, 33, 36, 47ovmpod 7556 1 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ (𝑀 Ismty 𝑁) = {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  Vcvv 3474   βŠ† wss 3947  βˆͺ cuni 4907   Γ— cxp 5673  dom cdm 5675  ran crn 5676  βŸΆwf 6536  β€“1-1-ontoβ†’wf1o 6539  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ↑m cmap 8816  β„*cxr 11243  βˆžMetcxmet 20921   Ismty cismty 36654
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-cnex 11162  ax-resscn 11163
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8818  df-xr 11248  df-xmet 20929  df-ismty 36655
This theorem is referenced by:  isismty  36657
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