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Theorem ismtyima 34962
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 5933 . . . . 5 (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2 isismty 34960 . . . . . . . . . 10 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
32biimp3a 1460 . . . . . . . . 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 6608 . . . . . . 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 6514 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ran 𝐹𝑌)
91, 8sstrid 3975 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
109sseld 3963 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) → 𝑥𝑌))
11 simpl2 1184 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌))
12 simprl 767 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑃𝑋)
13 ffvelrn 6841 . . . . . 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 769 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑅 ∈ ℝ*)
16 blssm 22955 . . . . 5 ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑅 ∈ ℝ*) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1711, 14, 15, 16syl3anc 1363 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1817sseld 3963 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) → 𝑥𝑌))
19 simpl1 1183 . . . . . . . . 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 774 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑅 ∈ ℝ*)
22 simplrl 773 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑃𝑋)
23 f1ocnv 6620 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
24 f1of 6608 . . . . . . . . . 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 ffvelrn 6841 . . . . . . . . 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 22927 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋 ∧ (𝐹𝑥) ∈ 𝑋)) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(𝐹𝑥)) < 𝑅))
2920, 21, 22, 27, 28syl22anc 834 . . . . . . 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 7152 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
32 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑥 = 𝑃 → (𝐹𝑥) = (𝐹𝑃))
3332oveq1d 7160 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → ((𝐹𝑥)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹𝑦)))
3431, 33eqeq12d 2834 . . . . . . . . . . . . 13 (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦))))
35 oveq2 7153 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(𝐹𝑥)))
36 fveq2 6663 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑥) → (𝐹𝑦) = (𝐹‘(𝐹𝑥)))
3736oveq2d 7161 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → ((𝐹𝑃)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
3835, 37eqeq12d 2834 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑥) → ((𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦)) ↔ (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
3934, 38rspc2v 3630 . . . . . . . . . . . 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 5067 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝑃𝑀(𝐹𝑥)) < 𝑅 ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
4529, 44bitrd 280 . . . . . 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 6607 . . . . . . . . 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 22955 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5019, 12, 15, 49syl3anc 1363 . . . . . . . 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 7012 . . . . . . 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 1363 . . . . . 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 7025 . . . . . . . . 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 2910 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) ∈ 𝑌)
60 elbl2 22927 . . . . . . 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 834 . . . . . 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 312 . . . . 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 2894 . . . . 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 2894 . . . . 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 310 . . . 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 381 . 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 2816 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wss 3933   class class class wbr 5057  ccnv 5547  ran crn 5549  cima 5551  wf 6344  1-1wf1 6345  1-1-ontowf1o 6347  cfv 6348  (class class class)co 7145  *cxr 10662   < clt 10663  ∞Metcxmet 20458  ballcbl 20460   Ismty cismty 34957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-map 8397  df-xr 10667  df-psmet 20465  df-xmet 20466  df-bl 20468  df-ismty 34958
This theorem is referenced by:  ismtyhmeolem  34963  ismtybndlem  34965
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