Theorem ismtyima 38717

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 6196	. . . . 5 ⊢ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2		isismty 38715	. . . . . . . . . 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 1498	. . . . . . . . 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 486	. . . . . . . 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 500	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝐹:𝑋–1-1-onto→𝑌)
6		f1of 6822	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
7	5, 6	syl 18	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝐹:𝑋⟶𝑌)
8	7	frnd 6716	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ran 𝐹 ⊆ 𝑌)
9	1, 8	sstrid 3942	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
10	9	sseld 3930	. . 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 1211	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌))
12		simprl 783	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑃 ∈ 𝑋)
13		ffvelcdm 7079	. . . . . 6 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑃 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝑌)
14	7, 12, 13	syl2anc 596	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹‘𝑃) ∈ 𝑌)
15		simprr 785	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑅 ∈ ℝ*)
16		blssm 24730	. . . . 5 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑅 ∈ ℝ*) → ((𝐹‘𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
17	11, 14, 15, 16	syl3anc 1398	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ((𝐹‘𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
18	17	sseld 3930	. . 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 1210	. . . . . . . . 9 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋))
20	19	adantr 486	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑀 ∈ (∞Met‘𝑋))
21		simplrr 790	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ ℝ*)
22		simplrl 789	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑃 ∈ 𝑋)
23		f1ocnv 6835	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
24		f1of 6822	. . . . . . . . . 10 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
25	5, 23, 24	3syl 19	. . . . . . . . 9 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ◡𝐹:𝑌⟶𝑋)
26		ffvelcdm 7079	. . . . . . . . 9 ⊢ ((◡𝐹:𝑌⟶𝑋 ∧ 𝑥 ∈ 𝑌) → (◡𝐹‘𝑥) ∈ 𝑋)
27	25, 26	sylan 592	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹‘𝑥) ∈ 𝑋)
28		elbl2 24702	. . . . . . . 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 852	. . . . . . 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 501	. . . . . . . . . . 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 7425	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
32		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑃 → (𝐹‘𝑥) = (𝐹‘𝑃))
33	32	oveq1d 7433	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)))
34	31, 33	eqeq12d 2777	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦))))
35		oveq2 7426	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡𝐹‘𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(◡𝐹‘𝑥)))
36		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (◡𝐹‘𝑥) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑥)))
37	36	oveq2d 7434	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡𝐹‘𝑥) → ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))))
38	35, 37	eqeq12d 2777	. . . . . . . . . . . . 13 ⊢ (𝑦 = (◡𝐹‘𝑥) → ((𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀(◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥)))))
39	34, 38	rspc2v 3587	. . . . . . . . . . . 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 457	. . . . . . . . . . 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 596	. . . . . . . . . 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 412	. . . . . . . . 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 603	. . . . . . . 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 5113	. . . . . . 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 282	. . . . . 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 6821	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
47	5, 46	syl 18	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝐹:𝑋–1-1→𝑌)
48	47	adantr 486	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋–1-1→𝑌)
49		blssm 24730	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
50	19, 12, 15, 49	syl3anc 1398	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
51	50	adantr 486	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
52		f1elima 7265	. . . . . . 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 1398	. . . . . 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 486	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑁 ∈ (∞Met‘𝑌))
55	14	adantr 486	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘𝑃) ∈ 𝑌)
56		f1ocnvfv2 7283	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
57	5, 56	sylan 592	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
58		simpr 490	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
59	57, 58	eqeltrd 2861	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) ∈ 𝑌)
60		elbl2 24702	. . . . . . 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 852	. . . . . 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 314	. . . . 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 2846	. . . . 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 2846	. . . . 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 312	. . . 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 418	. . 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 382	. 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 2759	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))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899   class class class wbr 5103  ^◡ccnv 5650  ran crn 5652   “ cima 5654  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7418  ℝ^*cxr 11335   < clt 11336  ∞Metcxmet 21656  ballcbl 21658   Ismty cismty 38712

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-map 8842  df-xr 11340  df-psmet 21663  df-xmet 21664  df-bl 21666  df-ismty 38713

This theorem is used by:  ismtyhmeolem  38718  ismtybndlem  38720