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Theorem ismtyres 36676
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 36669 . . . . . 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 500 . . . . 5 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
32adantrr 716 . . . 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 6833 . . . 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 772 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐴 βŠ† 𝑋)
7 f1ores 6848 . . 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 585 . 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 478 . . . 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 716 . . 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 3976 . . . . . . . . . . . . 13 (𝐴 βŠ† 𝑋 β†’ (𝑒 ∈ 𝐴 β†’ 𝑒 ∈ 𝑋))
12 ssel 3976 . . . . . . . . . . . . 13 (𝐴 βŠ† 𝑋 β†’ (𝑣 ∈ 𝐴 β†’ 𝑣 ∈ 𝑋))
1311, 12anim12d 610 . . . . . . . . . . . 12 (𝐴 βŠ† 𝑋 β†’ ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)))
1413imp 408 . . . . . . . . . . 11 ((𝐴 βŠ† 𝑋 ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋))
15 oveq1 7416 . . . . . . . . . . . . 13 (π‘₯ = 𝑒 β†’ (π‘₯𝑀𝑦) = (𝑒𝑀𝑦))
16 fveq2 6892 . . . . . . . . . . . . . 14 (π‘₯ = 𝑒 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘’))
1716oveq1d 7424 . . . . . . . . . . . . 13 (π‘₯ = 𝑒 β†’ ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)))
1815, 17eqeq12d 2749 . . . . . . . . . . . 12 (π‘₯ = 𝑒 β†’ ((π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)) ↔ (𝑒𝑀𝑦) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦))))
19 oveq2 7417 . . . . . . . . . . . . 13 (𝑦 = 𝑣 β†’ (𝑒𝑀𝑦) = (𝑒𝑀𝑣))
20 fveq2 6892 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘£))
2120oveq2d 7425 . . . . . . . . . . . . 13 (𝑦 = 𝑣 β†’ ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
2219, 21eqeq12d 2749 . . . . . . . . . . . 12 (𝑦 = 𝑣 β†’ ((𝑒𝑀𝑦) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)) ↔ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£))))
2318, 22rspc2v 3623 . . . . . . . . . . 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 408 . . . . . . . . 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 651 . . . . . . . 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 719 . . . . . . 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 717 . . . . . 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 7422 . . . . . . . 8 (𝑒𝑆𝑣) = (𝑒(𝑀 β†Ύ (𝐴 Γ— 𝐴))𝑣)
31 ovres 7573 . . . . . . . 8 ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒(𝑀 β†Ύ (𝐴 Γ— 𝐴))𝑣) = (𝑒𝑀𝑣))
3230, 31eqtrid 2785 . . . . . . 7 ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒𝑆𝑣) = (𝑒𝑀𝑣))
3332adantl 483 . . . . . 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 6911 . . . . . . . . . . 11 (𝑒 ∈ 𝐴 β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘’) = (πΉβ€˜π‘’))
3534ad2antrl 727 . . . . . . . . . 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 6911 . . . . . . . . . . 11 (𝑣 ∈ 𝐴 β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘£) = (πΉβ€˜π‘£))
3736ad2antll 728 . . . . . . . . . 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 7427 . . . . . . . . 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 7422 . . . . . . . . . 10 ((πΉβ€˜π‘’)𝑇(πΉβ€˜π‘£)) = ((πΉβ€˜π‘’)(𝑁 β†Ύ (𝐡 Γ— 𝐡))(πΉβ€˜π‘£))
41 f1ofun 6836 . . . . . . . . . . . . . . . 16 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ Fun 𝐹)
4241adantl 483 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ Fun 𝐹)
43 f1odm 6838 . . . . . . . . . . . . . . . . 17 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ dom 𝐹 = 𝑋)
4443sseq2d 4015 . . . . . . . . . . . . . . . 16 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ (𝐴 βŠ† dom 𝐹 ↔ 𝐴 βŠ† 𝑋))
4544biimparc 481 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ 𝐴 βŠ† dom 𝐹)
46 funfvima2 7233 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ 𝐴 βŠ† dom 𝐹) β†’ (𝑒 ∈ 𝐴 β†’ (πΉβ€˜π‘’) ∈ (𝐹 β€œ 𝐴)))
4742, 45, 46syl2anc 585 . . . . . . . . . . . . . 14 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (𝑒 ∈ 𝐴 β†’ (πΉβ€˜π‘’) ∈ (𝐹 β€œ 𝐴)))
4847imp 408 . . . . . . . . . . . . 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 2845 . . . . . . . . . . . 12 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑒 ∈ 𝐴) β†’ (πΉβ€˜π‘’) ∈ 𝐡)
5150adantrr 716 . . . . . . . . . . 11 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (πΉβ€˜π‘’) ∈ 𝐡)
52 funfvima2 7233 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ 𝐴 βŠ† dom 𝐹) β†’ (𝑣 ∈ 𝐴 β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴)))
5342, 45, 52syl2anc 585 . . . . . . . . . . . . . 14 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (𝑣 ∈ 𝐴 β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴)))
5453imp 408 . . . . . . . . . . . . 13 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑣 ∈ 𝐴) β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴))
5554, 49eleqtrrdi 2845 . . . . . . . . . . . 12 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑣 ∈ 𝐴) β†’ (πΉβ€˜π‘£) ∈ 𝐡)
5655adantrl 715 . . . . . . . . . . 11 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (πΉβ€˜π‘£) ∈ 𝐡)
5751, 56ovresd 7574 . . . . . . . . . 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 2785 . . . . . . . . 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 2773 . . . . . . . 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 720 . . . . . . 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 717 . . . . . 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 2783 . . . . 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 3201 . . . 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 719 . . 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 686 . 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 23867 . . . . 5 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑀 β†Ύ (𝐴 Γ— 𝐴)) ∈ (∞Metβ€˜π΄))
6729, 66eqeltrid 2838 . . . 4 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝑆 ∈ (∞Metβ€˜π΄))
6867ad2ant2rl 748 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑆 ∈ (∞Metβ€˜π΄))
69 simplr 768 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))
70 imassrn 6071 . . . . . . . 8 (𝐹 β€œ 𝐴) βŠ† ran 𝐹
7149, 70eqsstri 4017 . . . . . . 7 𝐡 βŠ† ran 𝐹
72 f1ofo 6841 . . . . . . . 8 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
73 forn 6809 . . . . . . . 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 4035 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐡 βŠ† π‘Œ)
76 xmetres2 23867 . . . . . 6 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ 𝐡 βŠ† π‘Œ) β†’ (𝑁 β†Ύ (𝐡 Γ— 𝐡)) ∈ (∞Metβ€˜π΅))
7769, 75, 76syl2anc 585 . . . . 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 2838 . . . 4 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑇 ∈ (∞Metβ€˜π΅))
7949fveq2i 6895 . . . 4 (∞Metβ€˜π΅) = (∞Metβ€˜(𝐹 β€œ 𝐴))
8078, 79eleqtrdi 2844 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑇 ∈ (∞Metβ€˜(𝐹 β€œ 𝐴)))
81 isismty 36669 . . 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 585 . 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 712 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062   βŠ† wss 3949   Γ— cxp 5675  dom cdm 5677  ran crn 5678   β†Ύ cres 5679   β€œ cima 5680  Fun wfun 6538  β€“1-1β†’wf1 6541  β€“ontoβ†’wfo 6542  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-res 5689  df-ima 5690  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:  reheibor  36707
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