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Theorem ismtyima 36262
Description: The image of a ball under an isometry is another ball. (Contributed by Jeff Madsen, 31-Jan-2014.)
Assertion
Ref Expression
ismtyima (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) = ((𝐹𝑃)(ball‘𝑁)𝑅))

Proof of Theorem ismtyima
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassrn 6024 . . . . 5 (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2 isismty 36260 . . . . . . . . . 10 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
32biimp3a 1469 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))))
43adantr 481 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))))
54simpld 495 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋1-1-onto𝑌)
6 f1of 6784 . . . . . . 7 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋𝑌)
75, 6syl 17 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋𝑌)
87frnd 6676 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ran 𝐹𝑌)
91, 8sstrid 3955 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
109sseld 3943 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) → 𝑥𝑌))
11 simpl2 1192 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌))
12 simprl 769 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑃𝑋)
13 ffvelcdm 7032 . . . . . 6 ((𝐹:𝑋𝑌𝑃𝑋) → (𝐹𝑃) ∈ 𝑌)
147, 12, 13syl2anc 584 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹𝑃) ∈ 𝑌)
15 simprr 771 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑅 ∈ ℝ*)
16 blssm 23771 . . . . 5 ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑅 ∈ ℝ*) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1711, 14, 15, 16syl3anc 1371 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1817sseld 3943 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) → 𝑥𝑌))
19 simpl1 1191 . . . . . . . . 9 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋))
2019adantr 481 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑀 ∈ (∞Met‘𝑋))
21 simplrr 776 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑅 ∈ ℝ*)
22 simplrl 775 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑃𝑋)
23 f1ocnv 6796 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
24 f1of 6784 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
255, 23, 243syl 18 . . . . . . . . 9 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑌𝑋)
26 ffvelcdm 7032 . . . . . . . . 9 ((𝐹:𝑌𝑋𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
2725, 26sylan 580 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
28 elbl2 23743 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋 ∧ (𝐹𝑥) ∈ 𝑋)) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(𝐹𝑥)) < 𝑅))
2920, 21, 22, 27, 28syl22anc 837 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(𝐹𝑥)) < 𝑅))
304simprd 496 . . . . . . . . . . 11 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))
31 oveq1 7364 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
32 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑥 = 𝑃 → (𝐹𝑥) = (𝐹𝑃))
3332oveq1d 7372 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → ((𝐹𝑥)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹𝑦)))
3431, 33eqeq12d 2752 . . . . . . . . . . . . 13 (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦))))
35 oveq2 7365 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(𝐹𝑥)))
36 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑥) → (𝐹𝑦) = (𝐹‘(𝐹𝑥)))
3736oveq2d 7373 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → ((𝐹𝑃)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
3835, 37eqeq12d 2752 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑥) → ((𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦)) ↔ (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
3934, 38rspc2v 3590 . . . . . . . . . . . 12 ((𝑃𝑋 ∧ (𝐹𝑥) ∈ 𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4039impancom 452 . . . . . . . . . . 11 ((𝑃𝑋 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ((𝐹𝑥) ∈ 𝑋 → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4112, 30, 40syl2anc 584 . . . . . . . . . 10 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑥) ∈ 𝑋 → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4241imp 407 . . . . . . . . 9 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ (𝐹𝑥) ∈ 𝑋) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
4327, 42syldan 591 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
4443breq1d 5115 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝑃𝑀(𝐹𝑥)) < 𝑅 ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
4529, 44bitrd 278 . . . . . 6 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
46 f1of1 6783 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋1-1𝑌)
475, 46syl 17 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋1-1𝑌)
4847adantr 481 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝐹:𝑋1-1𝑌)
49 blssm 23771 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5019, 12, 15, 49syl3anc 1371 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5150adantr 481 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
52 f1elima 7210 . . . . . . 7 ((𝐹:𝑋1-1𝑌 ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅)))
5348, 27, 51, 52syl3anc 1371 . . . . . 6 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅)))
5411adantr 481 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑁 ∈ (∞Met‘𝑌))
5514adantr 481 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹𝑃) ∈ 𝑌)
56 f1ocnvfv2 7223 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
575, 56sylan 580 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
58 simpr 485 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑥𝑌)
5957, 58eqeltrd 2838 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) ∈ 𝑌)
60 elbl2 23743 . . . . . . 7 (((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑅 ∈ ℝ*) ∧ ((𝐹𝑃) ∈ 𝑌 ∧ (𝐹‘(𝐹𝑥)) ∈ 𝑌)) → ((𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
6154, 21, 55, 59, 60syl22anc 837 . . . . . 6 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
6245, 53, 613bitr4d 310 . . . . 5 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6357eleq1d 2822 . . . . 5 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅))))
6457eleq1d 2822 . . . . 5 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6562, 63, 643bitr3d 308 . . . 4 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6665ex 413 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥𝑌 → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅))))
6710, 18, 66pm5.21ndd 380 . 2 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6867eqrdv 2734 1 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) = ((𝐹𝑃)(ball‘𝑁)𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wss 3910   class class class wbr 5105  ccnv 5632  ran crn 5634  cima 5636  wf 6492  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  *cxr 11188   < clt 11189  ∞Metcxmet 20781  ballcbl 20783   Ismty cismty 36257
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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767  df-xr 11193  df-psmet 20788  df-xmet 20789  df-bl 20791  df-ismty 36258
This theorem is referenced by:  ismtyhmeolem  36263  ismtybndlem  36265
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