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Theorem ismtyres 36979
Description: A restriction of an isometry is an isometry. The condition 𝐴 βŠ† 𝑋 is not necessary but makes the proof easier. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)
Hypotheses
Ref Expression
ismtyres.2 𝐡 = (𝐹 β€œ 𝐴)
ismtyres.3 𝑆 = (𝑀 β†Ύ (𝐴 Γ— 𝐴))
ismtyres.4 𝑇 = (𝑁 β†Ύ (𝐡 Γ— 𝐡))
Assertion
Ref Expression
ismtyres (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ (𝐹 β†Ύ 𝐴) ∈ (𝑆 Ismty 𝑇))

Proof of Theorem ismtyres
Dummy variables 𝑣 𝑒 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isismty 36972 . . . . . 6 ((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‘¦)))))
21simprbda 497 . . . . 5 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
32adantrr 713 . . . 4 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
4 f1of1 6831 . . . 4 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝐹:𝑋–1-1β†’π‘Œ)
53, 4syl 17 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐹:𝑋–1-1β†’π‘Œ)
6 simprr 769 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐴 βŠ† 𝑋)
7 f1ores 6846 . . 3 ((𝐹:𝑋–1-1β†’π‘Œ ∧ 𝐴 βŠ† 𝑋) β†’ (𝐹 β†Ύ 𝐴):𝐴–1-1-ontoβ†’(𝐹 β€œ 𝐴))
85, 6, 7syl2anc 582 . 2 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ (𝐹 β†Ύ 𝐴):𝐴–1-1-ontoβ†’(𝐹 β€œ 𝐴))
91biimpa 475 . . . 4 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))))
109adantrr 713 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))))
11 ssel 3974 . . . . . . . . . . . . 13 (𝐴 βŠ† 𝑋 β†’ (𝑒 ∈ 𝐴 β†’ 𝑒 ∈ 𝑋))
12 ssel 3974 . . . . . . . . . . . . 13 (𝐴 βŠ† 𝑋 β†’ (𝑣 ∈ 𝐴 β†’ 𝑣 ∈ 𝑋))
1311, 12anim12d 607 . . . . . . . . . . . 12 (𝐴 βŠ† 𝑋 β†’ ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)))
1413imp 405 . . . . . . . . . . 11 ((𝐴 βŠ† 𝑋 ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋))
15 oveq1 7418 . . . . . . . . . . . . 13 (π‘₯ = 𝑒 β†’ (π‘₯𝑀𝑦) = (𝑒𝑀𝑦))
16 fveq2 6890 . . . . . . . . . . . . . 14 (π‘₯ = 𝑒 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘’))
1716oveq1d 7426 . . . . . . . . . . . . 13 (π‘₯ = 𝑒 β†’ ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)))
1815, 17eqeq12d 2746 . . . . . . . . . . . 12 (π‘₯ = 𝑒 β†’ ((π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)) ↔ (𝑒𝑀𝑦) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦))))
19 oveq2 7419 . . . . . . . . . . . . 13 (𝑦 = 𝑣 β†’ (𝑒𝑀𝑦) = (𝑒𝑀𝑣))
20 fveq2 6890 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘£))
2120oveq2d 7427 . . . . . . . . . . . . 13 (𝑦 = 𝑣 β†’ ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
2219, 21eqeq12d 2746 . . . . . . . . . . . 12 (𝑦 = 𝑣 β†’ ((𝑒𝑀𝑦) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)) ↔ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£))))
2318, 22rspc2v 3621 . . . . . . . . . . 11 ((𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)) β†’ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£))))
2414, 23syl 17 . . . . . . . . . 10 ((𝐴 βŠ† 𝑋 ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)) β†’ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£))))
2524imp 405 . . . . . . . . 9 (((𝐴 βŠ† 𝑋 ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))) β†’ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
2625an32s 648 . . . . . . . 8 (((𝐴 βŠ† 𝑋 ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦))) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
2726adantlrl 716 . . . . . . 7 (((𝐴 βŠ† 𝑋 ∧ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
2827adantlll 714 . . . . . 6 (((((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐴 βŠ† 𝑋) ∧ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
29 ismtyres.3 . . . . . . . . 9 𝑆 = (𝑀 β†Ύ (𝐴 Γ— 𝐴))
3029oveqi 7424 . . . . . . . 8 (𝑒𝑆𝑣) = (𝑒(𝑀 β†Ύ (𝐴 Γ— 𝐴))𝑣)
31 ovres 7575 . . . . . . . 8 ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒(𝑀 β†Ύ (𝐴 Γ— 𝐴))𝑣) = (𝑒𝑀𝑣))
3230, 31eqtrid 2782 . . . . . . 7 ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒𝑆𝑣) = (𝑒𝑀𝑣))
3332adantl 480 . . . . . 6 (((((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐴 βŠ† 𝑋) ∧ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒𝑆𝑣) = (𝑒𝑀𝑣))
34 fvres 6909 . . . . . . . . . . 11 (𝑒 ∈ 𝐴 β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘’) = (πΉβ€˜π‘’))
3534ad2antrl 724 . . . . . . . . . 10 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘’) = (πΉβ€˜π‘’))
36 fvres 6909 . . . . . . . . . . 11 (𝑣 ∈ 𝐴 β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘£) = (πΉβ€˜π‘£))
3736ad2antll 725 . . . . . . . . . 10 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘£) = (πΉβ€˜π‘£))
3835, 37oveq12d 7429 . . . . . . . . 9 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)) = ((πΉβ€˜π‘’)𝑇(πΉβ€˜π‘£)))
39 ismtyres.4 . . . . . . . . . . 11 𝑇 = (𝑁 β†Ύ (𝐡 Γ— 𝐡))
4039oveqi 7424 . . . . . . . . . 10 ((πΉβ€˜π‘’)𝑇(πΉβ€˜π‘£)) = ((πΉβ€˜π‘’)(𝑁 β†Ύ (𝐡 Γ— 𝐡))(πΉβ€˜π‘£))
41 f1ofun 6834 . . . . . . . . . . . . . . . 16 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ Fun 𝐹)
4241adantl 480 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ Fun 𝐹)
43 f1odm 6836 . . . . . . . . . . . . . . . . 17 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ dom 𝐹 = 𝑋)
4443sseq2d 4013 . . . . . . . . . . . . . . . 16 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ (𝐴 βŠ† dom 𝐹 ↔ 𝐴 βŠ† 𝑋))
4544biimparc 478 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ 𝐴 βŠ† dom 𝐹)
46 funfvima2 7234 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ 𝐴 βŠ† dom 𝐹) β†’ (𝑒 ∈ 𝐴 β†’ (πΉβ€˜π‘’) ∈ (𝐹 β€œ 𝐴)))
4742, 45, 46syl2anc 582 . . . . . . . . . . . . . 14 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (𝑒 ∈ 𝐴 β†’ (πΉβ€˜π‘’) ∈ (𝐹 β€œ 𝐴)))
4847imp 405 . . . . . . . . . . . . 13 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑒 ∈ 𝐴) β†’ (πΉβ€˜π‘’) ∈ (𝐹 β€œ 𝐴))
49 ismtyres.2 . . . . . . . . . . . . 13 𝐡 = (𝐹 β€œ 𝐴)
5048, 49eleqtrrdi 2842 . . . . . . . . . . . 12 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑒 ∈ 𝐴) β†’ (πΉβ€˜π‘’) ∈ 𝐡)
5150adantrr 713 . . . . . . . . . . 11 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (πΉβ€˜π‘’) ∈ 𝐡)
52 funfvima2 7234 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ 𝐴 βŠ† dom 𝐹) β†’ (𝑣 ∈ 𝐴 β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴)))
5342, 45, 52syl2anc 582 . . . . . . . . . . . . . 14 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (𝑣 ∈ 𝐴 β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴)))
5453imp 405 . . . . . . . . . . . . 13 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑣 ∈ 𝐴) β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴))
5554, 49eleqtrrdi 2842 . . . . . . . . . . . 12 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑣 ∈ 𝐴) β†’ (πΉβ€˜π‘£) ∈ 𝐡)
5655adantrl 712 . . . . . . . . . . 11 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (πΉβ€˜π‘£) ∈ 𝐡)
5751, 56ovresd 7576 . . . . . . . . . 10 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ ((πΉβ€˜π‘’)(𝑁 β†Ύ (𝐡 Γ— 𝐡))(πΉβ€˜π‘£)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
5840, 57eqtrid 2782 . . . . . . . . 9 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ ((πΉβ€˜π‘’)𝑇(πΉβ€˜π‘£)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
5938, 58eqtrd 2770 . . . . . . . 8 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
6059adantlrr 717 . . . . . . 7 (((𝐴 βŠ† 𝑋 ∧ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
6160adantlll 714 . . . . . 6 (((((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐴 βŠ† 𝑋) ∧ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
6228, 33, 613eqtr4d 2780 . . . . 5 (((((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐴 βŠ† 𝑋) ∧ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒𝑆𝑣) = (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)))
6362ralrimivva 3198 . . . 4 ((((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐴 βŠ† 𝑋) ∧ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))) β†’ βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐴 (𝑒𝑆𝑣) = (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)))
6463adantlrl 716 . . 3 ((((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) ∧ (𝐹:𝑋–1-1-ontoβ†’π‘Œ ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)))) β†’ βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐴 (𝑒𝑆𝑣) = (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)))
6510, 64mpdan 683 . 2 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐴 (𝑒𝑆𝑣) = (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)))
66 xmetres2 24087 . . . . 5 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑀 β†Ύ (𝐴 Γ— 𝐴)) ∈ (∞Metβ€˜π΄))
6729, 66eqeltrid 2835 . . . 4 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝑆 ∈ (∞Metβ€˜π΄))
6867ad2ant2rl 745 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑆 ∈ (∞Metβ€˜π΄))
69 simplr 765 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))
70 imassrn 6069 . . . . . . . 8 (𝐹 β€œ 𝐴) βŠ† ran 𝐹
7149, 70eqsstri 4015 . . . . . . 7 𝐡 βŠ† ran 𝐹
72 f1ofo 6839 . . . . . . . 8 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
73 forn 6807 . . . . . . . 8 (𝐹:𝑋–ontoβ†’π‘Œ β†’ ran 𝐹 = π‘Œ)
743, 72, 733syl 18 . . . . . . 7 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ ran 𝐹 = π‘Œ)
7571, 74sseqtrid 4033 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐡 βŠ† π‘Œ)
76 xmetres2 24087 . . . . . 6 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ 𝐡 βŠ† π‘Œ) β†’ (𝑁 β†Ύ (𝐡 Γ— 𝐡)) ∈ (∞Metβ€˜π΅))
7769, 75, 76syl2anc 582 . . . . 5 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ (𝑁 β†Ύ (𝐡 Γ— 𝐡)) ∈ (∞Metβ€˜π΅))
7839, 77eqeltrid 2835 . . . 4 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑇 ∈ (∞Metβ€˜π΅))
7949fveq2i 6893 . . . 4 (∞Metβ€˜π΅) = (∞Metβ€˜(𝐹 β€œ 𝐴))
8078, 79eleqtrdi 2841 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑇 ∈ (∞Metβ€˜(𝐹 β€œ 𝐴)))
81 isismty 36972 . . 3 ((𝑆 ∈ (∞Metβ€˜π΄) ∧ 𝑇 ∈ (∞Metβ€˜(𝐹 β€œ 𝐴))) β†’ ((𝐹 β†Ύ 𝐴) ∈ (𝑆 Ismty 𝑇) ↔ ((𝐹 β†Ύ 𝐴):𝐴–1-1-ontoβ†’(𝐹 β€œ 𝐴) ∧ βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐴 (𝑒𝑆𝑣) = (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)))))
8268, 80, 81syl2anc 582 . 2 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ ((𝐹 β†Ύ 𝐴) ∈ (𝑆 Ismty 𝑇) ↔ ((𝐹 β†Ύ 𝐴):𝐴–1-1-ontoβ†’(𝐹 β€œ 𝐴) ∧ βˆ€π‘’ ∈ 𝐴 βˆ€π‘£ ∈ 𝐴 (𝑒𝑆𝑣) = (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)))))
838, 65, 82mpbir2and 709 1 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ (𝐹 β†Ύ 𝐴) ∈ (𝑆 Ismty 𝑇))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059   βŠ† wss 3947   Γ— cxp 5673  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   β€œ cima 5678  Fun wfun 6536  β€“1-1β†’wf1 6539  β€“ontoβ†’wfo 6540  β€“1-1-ontoβ†’wf1o 6541  β€˜cfv 6542  (class class class)co 7411  βˆžMetcxmet 21129   Ismty cismty 36969
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  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-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-map 8824  df-xr 11256  df-xmet 21137  df-ismty 36970
This theorem is referenced by:  reheibor  37010
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