Theorem ismtyima 38124

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)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) = ((𝐹‘𝑃)(ball‘𝑁)𝑅))

Proof of Theorem ismtyima

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 6036	. . . . 5 ⊢ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2		isismty 38122	. . . . . . . . . 10 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))))
3	2	biimp3a 1472	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))
4	3	adantr 480	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))
5	4	simpld 494	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝐹:𝑋–1-1-onto→𝑌)
6		f1of 6780	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
7	5, 6	syl 17	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝐹:𝑋⟶𝑌)
8	7	frnd 6676	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ran 𝐹 ⊆ 𝑌)
9	1, 8	sstrid 3933	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
10	9	sseld 3920	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) → 𝑥 ∈ 𝑌))
11		simpl2 1194	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌))
12		simprl 771	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑃 ∈ 𝑋)
13		ffvelcdm 7033	. . . . . 6 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑃 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝑌)
14	7, 12, 13	syl2anc 585	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹‘𝑃) ∈ 𝑌)
15		simprr 773	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑅 ∈ ℝ*)
16		blssm 24383	. . . . 5 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑅 ∈ ℝ*) → ((𝐹‘𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
17	11, 14, 15, 16	syl3anc 1374	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ((𝐹‘𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
18	17	sseld 3920	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅) → 𝑥 ∈ 𝑌))
19		simpl1 1193	. . . . . . . . 9 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋))
20	19	adantr 480	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑀 ∈ (∞Met‘𝑋))
21		simplrr 778	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ ℝ*)
22		simplrl 777	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑃 ∈ 𝑋)
23		f1ocnv 6792	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
24		f1of 6780	. . . . . . . . . 10 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
25	5, 23, 24	3syl 18	. . . . . . . . 9 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ◡𝐹:𝑌⟶𝑋)
26		ffvelcdm 7033	. . . . . . . . 9 ⊢ ((◡𝐹:𝑌⟶𝑋 ∧ 𝑥 ∈ 𝑌) → (◡𝐹‘𝑥) ∈ 𝑋)
27	25, 26	sylan 581	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹‘𝑥) ∈ 𝑋)
28		elbl2 24355	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ (◡𝐹‘𝑥) ∈ 𝑋)) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(◡𝐹‘𝑥)) < 𝑅))
29	20, 21, 22, 27, 28	syl22anc 839	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(◡𝐹‘𝑥)) < 𝑅))
30	4	simprd 495	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))
31		oveq1 7374	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
32		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑃 → (𝐹‘𝑥) = (𝐹‘𝑃))
33	32	oveq1d 7382	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)))
34	31, 33	eqeq12d 2752	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦))))
35		oveq2 7375	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡𝐹‘𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(◡𝐹‘𝑥)))
36		fveq2 6840	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (◡𝐹‘𝑥) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑥)))
37	36	oveq2d 7383	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡𝐹‘𝑥) → ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))))
38	35, 37	eqeq12d 2752	. . . . . . . . . . . . 13 ⊢ (𝑦 = (◡𝐹‘𝑥) → ((𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀(◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥)))))
39	34, 38	rspc2v 3575	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ 𝑋 ∧ (◡𝐹‘𝑥) ∈ 𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) → (𝑃𝑀(◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥)))))
40	39	impancom 451	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ((◡𝐹‘𝑥) ∈ 𝑋 → (𝑃𝑀(◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥)))))
41	12, 30, 40	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ((◡𝐹‘𝑥) ∈ 𝑋 → (𝑃𝑀(◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥)))))
42	41	imp 406	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ (◡𝐹‘𝑥) ∈ 𝑋) → (𝑃𝑀(◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))))
43	27, 42	syldan 592	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝑃𝑀(◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))))
44	43	breq1d 5095	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → ((𝑃𝑀(◡𝐹‘𝑥)) < 𝑅 ↔ ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))) < 𝑅))
45	29, 44	bitrd 279	. . . . . 6 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))) < 𝑅))
46		f1of1 6779	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
47	5, 46	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝐹:𝑋–1-1→𝑌)
48	47	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋–1-1→𝑌)
49		blssm 24383	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
50	19, 12, 15, 49	syl3anc 1374	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
51	50	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
52		f1elima 7218	. . . . . . 7 ⊢ ((𝐹:𝑋–1-1→𝑌 ∧ (◡𝐹‘𝑥) ∈ 𝑋 ∧ (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋) → ((𝐹‘(◡𝐹‘𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝑀)𝑅)))
53	48, 27, 51, 52	syl3anc 1374	. . . . . 6 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → ((𝐹‘(◡𝐹‘𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝑀)𝑅)))
54	11	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑁 ∈ (∞Met‘𝑌))
55	14	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘𝑃) ∈ 𝑌)
56		f1ocnvfv2 7232	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
57	5, 56	sylan 581	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
58		simpr 484	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
59	57, 58	eqeltrd 2836	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) ∈ 𝑌)
60		elbl2 24355	. . . . . . 7 ⊢ (((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑅 ∈ ℝ*) ∧ ((𝐹‘𝑃) ∈ 𝑌 ∧ (𝐹‘(◡𝐹‘𝑥)) ∈ 𝑌)) → ((𝐹‘(◡𝐹‘𝑥)) ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅) ↔ ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))) < 𝑅))
61	54, 21, 55, 59, 60	syl22anc 839	. . . . . 6 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → ((𝐹‘(◡𝐹‘𝑥)) ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅) ↔ ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))) < 𝑅))
62	45, 53, 61	3bitr4d 311	. . . . 5 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → ((𝐹‘(◡𝐹‘𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (𝐹‘(◡𝐹‘𝑥)) ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅)))
63	57	eleq1d 2821	. . . . 5 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → ((𝐹‘(◡𝐹‘𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅))))
64	57	eleq1d 2821	. . . . 5 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → ((𝐹‘(◡𝐹‘𝑥)) ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅) ↔ 𝑥 ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅)))
65	62, 63, 64	3bitr3d 309	. . . 4 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅)))
66	65	ex 412	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑥 ∈ 𝑌 → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅))))
67	10, 18, 66	pm5.21ndd 379	. 2 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅)))
68	67	eqrdv 2734	1 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) = ((𝐹‘𝑃)(ball‘𝑁)𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889   class class class wbr 5085  ^◡ccnv 5630  ran crn 5632   “ cima 5634  ⟶wf 6494  –1-1→wf1 6495  –1-1-onto→wf1o 6497  ‘cfv 6498  (class class class)co 7367  ℝ^*cxr 11178   < clt 11179  ∞Metcxmet 21337  ballcbl 21339   Ismty cismty 38119

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-xr 11183  df-psmet 21344  df-xmet 21345  df-bl 21347  df-ismty 38120

This theorem is referenced by:  ismtyhmeolem  38125  ismtybndlem  38127