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Theorem isismty 36669
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 36668 . . 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 6834 . . . . . . 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 6929 . . . . . 6 (𝑀 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 ∈ dom ∞Met)
6 elfvdm 6929 . . . . . 6 (𝑁 ∈ (∞Metβ€˜π‘Œ) β†’ π‘Œ ∈ dom ∞Met)
7 fex2 7924 . . . . . 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 6822 . . . . 5 (𝑓 = 𝐹 β†’ (𝑓:𝑋–1-1-ontoβ†’π‘Œ ↔ 𝐹:𝑋–1-1-ontoβ†’π‘Œ))
12 fveq1 6891 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘₯) = (πΉβ€˜π‘₯))
13 fveq1 6891 . . . . . . . 8 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
1412, 13oveq12d 7427 . . . . . . 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 3219 . . . . 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 3676 . . 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 3062  Vcvv 3475  dom cdm 5677  βŸΆwf 6540  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409  βˆžMetcxmet 20929   Ismty cismty 36666
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-map 8822  df-xr 11252  df-xmet 20937  df-ismty 36667
This theorem is referenced by:  ismtycnv  36670  ismtyima  36671  ismtyhmeolem  36672  ismtybndlem  36674  ismtyres  36676  ismrer1  36706  reheibor  36707
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