Mathbox for Jeff Madsen < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ismtyima Structured version   Visualization version   GIF version

Theorem ismtyima 33255
 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 5438 . . . . 5 (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2 isismty 33253 . . . . . . . . . 10 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
32biimp3a 1429 . . . . . . . . 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 475 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋1-1-onto𝑌)
6 f1of 6096 . . . . . . 7 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋𝑌)
75, 6syl 17 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋𝑌)
8 frn 6012 . . . . . 6 (𝐹:𝑋𝑌 → ran 𝐹𝑌)
97, 8syl 17 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ran 𝐹𝑌)
101, 9syl5ss 3595 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
1110sseld 3583 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) → 𝑥𝑌))
12 simpl2 1063 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌))
13 simprl 793 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑃𝑋)
14 ffvelrn 6315 . . . . . 6 ((𝐹:𝑋𝑌𝑃𝑋) → (𝐹𝑃) ∈ 𝑌)
157, 13, 14syl2anc 692 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹𝑃) ∈ 𝑌)
16 simprr 795 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑅 ∈ ℝ*)
17 blssm 22136 . . . . 5 ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑅 ∈ ℝ*) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1812, 15, 16, 17syl3anc 1323 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1918sseld 3583 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) → 𝑥𝑌))
20 simpl1 1062 . . . . . . . . 9 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋))
2120adantr 481 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑀 ∈ (∞Met‘𝑋))
22 simplrr 800 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑅 ∈ ℝ*)
23 simplrl 799 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑃𝑋)
24 f1ocnv 6108 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
25 f1of 6096 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
265, 24, 253syl 18 . . . . . . . . 9 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑌𝑋)
27 ffvelrn 6315 . . . . . . . . 9 ((𝐹:𝑌𝑋𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
2826, 27sylan 488 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
29 elbl2 22108 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋 ∧ (𝐹𝑥) ∈ 𝑋)) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(𝐹𝑥)) < 𝑅))
3021, 22, 23, 28, 29syl22anc 1324 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(𝐹𝑥)) < 𝑅))
314simprd 479 . . . . . . . . . . 11 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))
32 oveq1 6614 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
33 fveq2 6150 . . . . . . . . . . . . . . 15 (𝑥 = 𝑃 → (𝐹𝑥) = (𝐹𝑃))
3433oveq1d 6622 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → ((𝐹𝑥)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹𝑦)))
3532, 34eqeq12d 2636 . . . . . . . . . . . . 13 (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦))))
36 oveq2 6615 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(𝐹𝑥)))
37 fveq2 6150 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑥) → (𝐹𝑦) = (𝐹‘(𝐹𝑥)))
3837oveq2d 6623 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → ((𝐹𝑃)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
3936, 38eqeq12d 2636 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑥) → ((𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦)) ↔ (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4035, 39rspc2v 3307 . . . . . . . . . . . 12 ((𝑃𝑋 ∧ (𝐹𝑥) ∈ 𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4140impancom 456 . . . . . . . . . . 11 ((𝑃𝑋 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ((𝐹𝑥) ∈ 𝑋 → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4213, 31, 41syl2anc 692 . . . . . . . . . 10 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑥) ∈ 𝑋 → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4342imp 445 . . . . . . . . 9 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ (𝐹𝑥) ∈ 𝑋) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
4428, 43syldan 487 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
4544breq1d 4625 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝑃𝑀(𝐹𝑥)) < 𝑅 ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
4630, 45bitrd 268 . . . . . 6 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
47 f1of1 6095 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋1-1𝑌)
485, 47syl 17 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋1-1𝑌)
4948adantr 481 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝐹:𝑋1-1𝑌)
50 blssm 22136 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5120, 13, 16, 50syl3anc 1323 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5251adantr 481 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
53 f1elima 6477 . . . . . . 7 ((𝐹:𝑋1-1𝑌 ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅)))
5449, 28, 52, 53syl3anc 1323 . . . . . 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)))
5512adantr 481 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑁 ∈ (∞Met‘𝑌))
5615adantr 481 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹𝑃) ∈ 𝑌)
57 f1ocnvfv2 6490 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
585, 57sylan 488 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
59 simpr 477 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑥𝑌)
6058, 59eqeltrd 2698 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) ∈ 𝑌)
61 elbl2 22108 . . . . . . 7 (((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑅 ∈ ℝ*) ∧ ((𝐹𝑃) ∈ 𝑌 ∧ (𝐹‘(𝐹𝑥)) ∈ 𝑌)) → ((𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
6255, 22, 56, 60, 61syl22anc 1324 . . . . . 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))
6346, 54, 623bitr4d 300 . . . . 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)))
6458eleq1d 2683 . . . . 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))))
6558eleq1d 2683 . . . . 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)))
6663, 64, 653bitr3d 298 . . . 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)))
6766ex 450 . . 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))))
6811, 19, 67pm5.21ndd 369 . 2 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6968eqrdv 2619 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 196   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907   ⊆ wss 3556   class class class wbr 4615  ◡ccnv 5075  ran crn 5077   “ cima 5079  ⟶wf 5845  –1-1→wf1 5846  –1-1-onto→wf1o 5848  ‘cfv 5849  (class class class)co 6607  ℝ*cxr 10020   < clt 10021  ∞Metcxmt 19653  ballcbl 19655   Ismty cismty 33250 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-1st 7116  df-2nd 7117  df-map 7807  df-xr 10025  df-psmet 19660  df-xmet 19661  df-bl 19663  df-ismty 33251 This theorem is referenced by:  ismtyhmeolem  33256  ismtybndlem  33258
 Copyright terms: Public domain W3C validator