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Theorem ismtyres 36671
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 36664 . . . . . 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 499 . . . . 5 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) β†’ 𝐹:𝑋–1-1-ontoβ†’π‘Œ)
32adantrr 715 . . . 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 6832 . . . 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 771 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐴 βŠ† 𝑋)
7 f1ores 6847 . . 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 584 . 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 477 . . . 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 715 . . 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 3975 . . . . . . . . . . . . 13 (𝐴 βŠ† 𝑋 β†’ (𝑒 ∈ 𝐴 β†’ 𝑒 ∈ 𝑋))
12 ssel 3975 . . . . . . . . . . . . 13 (𝐴 βŠ† 𝑋 β†’ (𝑣 ∈ 𝐴 β†’ 𝑣 ∈ 𝑋))
1311, 12anim12d 609 . . . . . . . . . . . 12 (𝐴 βŠ† 𝑋 β†’ ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋)))
1413imp 407 . . . . . . . . . . 11 ((𝐴 βŠ† 𝑋 ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (𝑒 ∈ 𝑋 ∧ 𝑣 ∈ 𝑋))
15 oveq1 7415 . . . . . . . . . . . . 13 (π‘₯ = 𝑒 β†’ (π‘₯𝑀𝑦) = (𝑒𝑀𝑦))
16 fveq2 6891 . . . . . . . . . . . . . 14 (π‘₯ = 𝑒 β†’ (πΉβ€˜π‘₯) = (πΉβ€˜π‘’))
1716oveq1d 7423 . . . . . . . . . . . . 13 (π‘₯ = 𝑒 β†’ ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)))
1815, 17eqeq12d 2748 . . . . . . . . . . . 12 (π‘₯ = 𝑒 β†’ ((π‘₯𝑀𝑦) = ((πΉβ€˜π‘₯)𝑁(πΉβ€˜π‘¦)) ↔ (𝑒𝑀𝑦) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦))))
19 oveq2 7416 . . . . . . . . . . . . 13 (𝑦 = 𝑣 β†’ (𝑒𝑀𝑦) = (𝑒𝑀𝑣))
20 fveq2 6891 . . . . . . . . . . . . . 14 (𝑦 = 𝑣 β†’ (πΉβ€˜π‘¦) = (πΉβ€˜π‘£))
2120oveq2d 7424 . . . . . . . . . . . . 13 (𝑦 = 𝑣 β†’ ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
2219, 21eqeq12d 2748 . . . . . . . . . . . 12 (𝑦 = 𝑣 β†’ ((𝑒𝑀𝑦) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘¦)) ↔ (𝑒𝑀𝑣) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£))))
2318, 22rspc2v 3622 . . . . . . . . . . 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 407 . . . . . . . . 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 650 . . . . . . . 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 718 . . . . . . 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 716 . . . . . 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 7421 . . . . . . . 8 (𝑒𝑆𝑣) = (𝑒(𝑀 β†Ύ (𝐴 Γ— 𝐴))𝑣)
31 ovres 7572 . . . . . . . 8 ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒(𝑀 β†Ύ (𝐴 Γ— 𝐴))𝑣) = (𝑒𝑀𝑣))
3230, 31eqtrid 2784 . . . . . . 7 ((𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴) β†’ (𝑒𝑆𝑣) = (𝑒𝑀𝑣))
3332adantl 482 . . . . . 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 6910 . . . . . . . . . . 11 (𝑒 ∈ 𝐴 β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘’) = (πΉβ€˜π‘’))
3534ad2antrl 726 . . . . . . . . . 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 6910 . . . . . . . . . . 11 (𝑣 ∈ 𝐴 β†’ ((𝐹 β†Ύ 𝐴)β€˜π‘£) = (πΉβ€˜π‘£))
3736ad2antll 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‘£))
3835, 37oveq12d 7426 . . . . . . . . 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 7421 . . . . . . . . . 10 ((πΉβ€˜π‘’)𝑇(πΉβ€˜π‘£)) = ((πΉβ€˜π‘’)(𝑁 β†Ύ (𝐡 Γ— 𝐡))(πΉβ€˜π‘£))
41 f1ofun 6835 . . . . . . . . . . . . . . . 16 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ Fun 𝐹)
4241adantl 482 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ Fun 𝐹)
43 f1odm 6837 . . . . . . . . . . . . . . . . 17 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ dom 𝐹 = 𝑋)
4443sseq2d 4014 . . . . . . . . . . . . . . . 16 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ (𝐴 βŠ† dom 𝐹 ↔ 𝐴 βŠ† 𝑋))
4544biimparc 480 . . . . . . . . . . . . . . 15 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ 𝐴 βŠ† dom 𝐹)
46 funfvima2 7232 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ 𝐴 βŠ† dom 𝐹) β†’ (𝑒 ∈ 𝐴 β†’ (πΉβ€˜π‘’) ∈ (𝐹 β€œ 𝐴)))
4742, 45, 46syl2anc 584 . . . . . . . . . . . . . 14 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (𝑒 ∈ 𝐴 β†’ (πΉβ€˜π‘’) ∈ (𝐹 β€œ 𝐴)))
4847imp 407 . . . . . . . . . . . . 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 2844 . . . . . . . . . . . 12 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑒 ∈ 𝐴) β†’ (πΉβ€˜π‘’) ∈ 𝐡)
5150adantrr 715 . . . . . . . . . . 11 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (πΉβ€˜π‘’) ∈ 𝐡)
52 funfvima2 7232 . . . . . . . . . . . . . . 15 ((Fun 𝐹 ∧ 𝐴 βŠ† dom 𝐹) β†’ (𝑣 ∈ 𝐴 β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴)))
5342, 45, 52syl2anc 584 . . . . . . . . . . . . . 14 ((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) β†’ (𝑣 ∈ 𝐴 β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴)))
5453imp 407 . . . . . . . . . . . . 13 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑣 ∈ 𝐴) β†’ (πΉβ€˜π‘£) ∈ (𝐹 β€œ 𝐴))
5554, 49eleqtrrdi 2844 . . . . . . . . . . . 12 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ 𝑣 ∈ 𝐴) β†’ (πΉβ€˜π‘£) ∈ 𝐡)
5655adantrl 714 . . . . . . . . . . 11 (((𝐴 βŠ† 𝑋 ∧ 𝐹:𝑋–1-1-ontoβ†’π‘Œ) ∧ (𝑒 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) β†’ (πΉβ€˜π‘£) ∈ 𝐡)
5751, 56ovresd 7573 . . . . . . . . . 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 2784 . . . . . . . . 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 2772 . . . . . . . 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 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†’ (((𝐹 β†Ύ 𝐴)β€˜π‘’)𝑇((𝐹 β†Ύ 𝐴)β€˜π‘£)) = ((πΉβ€˜π‘’)𝑁(πΉβ€˜π‘£)))
6160adantlll 716 . . . . . 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 2782 . . . . 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 3200 . . . 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 718 . . 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 685 . 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 23866 . . . . 5 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ (𝑀 β†Ύ (𝐴 Γ— 𝐴)) ∈ (∞Metβ€˜π΄))
6729, 66eqeltrid 2837 . . . 4 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝐴 βŠ† 𝑋) β†’ 𝑆 ∈ (∞Metβ€˜π΄))
6867ad2ant2rl 747 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑆 ∈ (∞Metβ€˜π΄))
69 simplr 767 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑁 ∈ (∞Metβ€˜π‘Œ))
70 imassrn 6070 . . . . . . . 8 (𝐹 β€œ 𝐴) βŠ† ran 𝐹
7149, 70eqsstri 4016 . . . . . . 7 𝐡 βŠ† ran 𝐹
72 f1ofo 6840 . . . . . . . 8 (𝐹:𝑋–1-1-ontoβ†’π‘Œ β†’ 𝐹:𝑋–ontoβ†’π‘Œ)
73 forn 6808 . . . . . . . 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 4034 . . . . . 6 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝐡 βŠ† π‘Œ)
76 xmetres2 23866 . . . . . 6 ((𝑁 ∈ (∞Metβ€˜π‘Œ) ∧ 𝐡 βŠ† π‘Œ) β†’ (𝑁 β†Ύ (𝐡 Γ— 𝐡)) ∈ (∞Metβ€˜π΅))
7769, 75, 76syl2anc 584 . . . . 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 2837 . . . 4 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑇 ∈ (∞Metβ€˜π΅))
7949fveq2i 6894 . . . 4 (∞Metβ€˜π΅) = (∞Metβ€˜(𝐹 β€œ 𝐴))
8078, 79eleqtrdi 2843 . . 3 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑁 ∈ (∞Metβ€˜π‘Œ)) ∧ (𝐹 ∈ (𝑀 Ismty 𝑁) ∧ 𝐴 βŠ† 𝑋)) β†’ 𝑇 ∈ (∞Metβ€˜(𝐹 β€œ 𝐴)))
81 isismty 36664 . . 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 584 . 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 711 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061   βŠ† wss 3948   Γ— cxp 5674  dom cdm 5676  ran crn 5677   β†Ύ cres 5678   β€œ cima 5679  Fun wfun 6537  β€“1-1β†’wf1 6540  β€“ontoβ†’wfo 6541  β€“1-1-ontoβ†’wf1o 6542  β€˜cfv 6543  (class class class)co 7408  βˆžMetcxmet 20928   Ismty cismty 36661
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 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166
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 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-xr 11251  df-xmet 20936  df-ismty 36662
This theorem is referenced by:  reheibor  36702
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