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Theorem isismty 36306
Description: The condition "is an isometry". (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
isismty ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))))
Distinct variable groups:   π‘₯,𝑀,𝑦   π‘₯,𝑁,𝑦   π‘₯,𝑋,𝑦   π‘₯,π‘Œ,𝑦   π‘₯,𝐹,𝑦

Proof of Theorem isismty
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 ismtyval 36305 . . 3 ((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‘¦)))})
21eleq2d 2820 . 2 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))}))
3 f1of 6785 . . . . . . 7 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝐹:π‘‹βŸΆπ‘Œ)
43adantr 482 . . . . . 6 ((𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))) β†’ 𝐹:π‘‹βŸΆπ‘Œ)
5 elfvdm 6880 . . . . . 6 (𝑀 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 ∈ dom ∞Met)
6 elfvdm 6880 . . . . . 6 (𝑁 ∈ (∞Metβ€˜π‘Œ) β†’ π‘Œ ∈ dom ∞Met)
7 fex2 7871 . . . . . 6 ((𝐹:π‘‹βŸΆπ‘Œ ∧ 𝑋 ∈ dom ∞Met ∧ π‘Œ ∈ dom ∞Met) β†’ 𝐹 ∈ V)
84, 5, 6, 7syl3an 1161 . . . . 5 (((𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))) ∧ 𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ 𝐹 ∈ V)
983expib 1123 . . . 4 ((𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))) β†’ ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ 𝐹 ∈ V))
109com12 32 . . 3 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ ((𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))) β†’ 𝐹 ∈ V))
11 f1oeq1 6773 . . . . 5 (𝑓 = 𝐹 β†’ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ↔ 𝐹:𝑋–1-1-ontoβ†’π‘Œ))
12 fveq1 6842 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
13 fveq1 6842 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
1412, 13oveq12d 7376 . . . . . . 7 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))
1514eqeq2d 2744 . . . . . 6 (𝑓 = 𝐹 β†’ ((π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)) ↔ (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))))
16152ralbidv 3209 . . . . 5 (𝑓 = 𝐹 β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)) ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))))
1711, 16anbi12d 632 . . . 4 (𝑓 = 𝐹 β†’ ((𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦))) ↔ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))))
1817elab3g 3638 . . 3 (((𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))) β†’ 𝐹 ∈ V) β†’ (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))} ↔ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))))
1910, 18syl 17 . 2 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((π‘“β€˜π‘₯)𝑁(π‘“β€˜π‘¦)))} ↔ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))))
202, 19bitrd 279 1 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) β†’ (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3444  dom cdm 5634  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358  βˆžMetcxmet 20797   Ismty cismty 36303
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-xr 11198  df-xmet 20805  df-ismty 36304
This theorem is referenced by:  ismtycnv  36307  ismtyima  36308  ismtyhmeolem  36309  ismtybndlem  36311  ismtyres  36313  ismrer1  36343  reheibor  36344
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