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Theorem ismtyima 35260
 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 5908 . . . . 5 (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2 isismty 35258 . . . . . . . . . 10 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
32biimp3a 1466 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))))
43adantr 484 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))))
54simpld 498 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋1-1-onto𝑌)
6 f1of 6591 . . . . . . 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 6495 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ran 𝐹𝑌)
91, 8sstrid 3926 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
109sseld 3914 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) → 𝑥𝑌))
11 simpl2 1189 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌))
12 simprl 770 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑃𝑋)
13 ffvelrn 6827 . . . . . 6 ((𝐹:𝑋𝑌𝑃𝑋) → (𝐹𝑃) ∈ 𝑌)
147, 12, 13syl2anc 587 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹𝑃) ∈ 𝑌)
15 simprr 772 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑅 ∈ ℝ*)
16 blssm 23035 . . . . 5 ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑅 ∈ ℝ*) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1711, 14, 15, 16syl3anc 1368 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1817sseld 3914 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) → 𝑥𝑌))
19 simpl1 1188 . . . . . . . . 9 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋))
2019adantr 484 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑀 ∈ (∞Met‘𝑋))
21 simplrr 777 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑅 ∈ ℝ*)
22 simplrl 776 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑃𝑋)
23 f1ocnv 6603 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
24 f1of 6591 . . . . . . . . . 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 6827 . . . . . . . . 9 ((𝐹:𝑌𝑋𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
2725, 26sylan 583 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
28 elbl2 23007 . . . . . . . 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 499 . . . . . . . . . . 11 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))
31 oveq1 7143 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
32 fveq2 6646 . . . . . . . . . . . . . . 15 (𝑥 = 𝑃 → (𝐹𝑥) = (𝐹𝑃))
3332oveq1d 7151 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → ((𝐹𝑥)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹𝑦)))
3431, 33eqeq12d 2814 . . . . . . . . . . . . 13 (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦))))
35 oveq2 7144 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(𝐹𝑥)))
36 fveq2 6646 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑥) → (𝐹𝑦) = (𝐹‘(𝐹𝑥)))
3736oveq2d 7152 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → ((𝐹𝑃)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
3835, 37eqeq12d 2814 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑥) → ((𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦)) ↔ (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
3934, 38rspc2v 3581 . . . . . . . . . . . 12 ((𝑃𝑋 ∧ (𝐹𝑥) ∈ 𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4039impancom 455 . . . . . . . . . . 11 ((𝑃𝑋 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ((𝐹𝑥) ∈ 𝑋 → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4112, 30, 40syl2anc 587 . . . . . . . . . 10 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑥) ∈ 𝑋 → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4241imp 410 . . . . . . . . 9 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ (𝐹𝑥) ∈ 𝑋) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
4327, 42syldan 594 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
4443breq1d 5041 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝑃𝑀(𝐹𝑥)) < 𝑅 ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
4529, 44bitrd 282 . . . . . 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 6590 . . . . . . . . 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 484 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝐹:𝑋1-1𝑌)
49 blssm 23035 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5019, 12, 15, 49syl3anc 1368 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5150adantr 484 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
52 f1elima 7000 . . . . . . 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 1368 . . . . . 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 484 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑁 ∈ (∞Met‘𝑌))
5514adantr 484 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹𝑃) ∈ 𝑌)
56 f1ocnvfv2 7013 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
575, 56sylan 583 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
58 simpr 488 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑥𝑌)
5957, 58eqeltrd 2890 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) ∈ 𝑌)
60 elbl2 23007 . . . . . . 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 314 . . . . 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 2874 . . . . 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 2874 . . . . 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 312 . . . 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 416 . . 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 384 . 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 2796 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881   class class class wbr 5031  ◡ccnv 5519  ran crn 5521   “ cima 5523  ⟶wf 6321  –1-1→wf1 6322  –1-1-onto→wf1o 6324  ‘cfv 6325  (class class class)co 7136  ℝ*cxr 10666   < clt 10667  ∞Metcxmet 20080  ballcbl 20082   Ismty cismty 35255 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-cnex 10585  ax-resscn 10586 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-map 8394  df-xr 10671  df-psmet 20087  df-xmet 20088  df-bl 20090  df-ismty 35256 This theorem is referenced by:  ismtyhmeolem  35261  ismtybndlem  35263
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