Theorem ismtyima 32772

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 5396	. . . . 5 ⊢ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2		isismty 32770	. . . . . . . . . 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 1424	. . . . . . . . 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 474	. . . . . . 7 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝐹:𝑋–1-1-onto→𝑌)
6		f1of 6050	. . . . . . 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		frn 5966	. . . . . 6 ⊢ (𝐹:𝑋⟶𝑌 → ran 𝐹 ⊆ 𝑌)
9	7, 8	syl 17	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ran 𝐹 ⊆ 𝑌)
10	1, 9	syl5ss 3579	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
11	10	sseld 3567	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) → 𝑥 ∈ 𝑌))
12		simpl2 1058	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌))
13		simprl 790	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑃 ∈ 𝑋)
14		ffvelrn 6265	. . . . . 6 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑃 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝑌)
15	7, 13, 14	syl2anc 691	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝐹‘𝑃) ∈ 𝑌)
16		simprr 792	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑅 ∈ ℝ*)
17		blssm 22033	. . . . 5 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑅 ∈ ℝ*) → ((𝐹‘𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
18	12, 15, 16, 17	syl3anc 1318	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ((𝐹‘𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
19	18	sseld 3567	. . 3 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝑁)𝑅) → 𝑥 ∈ 𝑌))
20		simpl1 1057	. . . . . . . . 9 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋))
21	20	adantr 480	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑀 ∈ (∞Met‘𝑋))
22		simplrr 797	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ ℝ*)
23		simplrl 796	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑃 ∈ 𝑋)
24		f1ocnv 6062	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋)
25		f1of 6050	. . . . . . . . . 10 ⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹:𝑌⟶𝑋)
26	5, 24, 25	3syl 18	. . . . . . . . 9 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → ◡𝐹:𝑌⟶𝑋)
27		ffvelrn 6265	. . . . . . . . 9 ⊢ ((◡𝐹:𝑌⟶𝑋 ∧ 𝑥 ∈ 𝑌) → (◡𝐹‘𝑥) ∈ 𝑋)
28	26, 27	sylan 487	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (◡𝐹‘𝑥) ∈ 𝑋)
29		elbl2 22005	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃 ∈ 𝑋 ∧ (◡𝐹‘𝑥) ∈ 𝑋)) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(◡𝐹‘𝑥)) < 𝑅))
30	21, 22, 23, 28, 29	syl22anc 1319	. . . . . . 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))
31	4	simprd 478	. . . . . . . . . . 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)))
32		oveq1 6556	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
33		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑃 → (𝐹‘𝑥) = (𝐹‘𝑃))
34	33	oveq1d 6564	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)))
35	32, 34	eqeq12d 2625	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦))))
36		oveq2 6557	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡𝐹‘𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(◡𝐹‘𝑥)))
37		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (◡𝐹‘𝑥) → (𝐹‘𝑦) = (𝐹‘(◡𝐹‘𝑥)))
38	37	oveq2d 6565	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (◡𝐹‘𝑥) → ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥))))
39	36, 38	eqeq12d 2625	. . . . . . . . . . . . 13 ⊢ (𝑦 = (◡𝐹‘𝑥) → ((𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀(◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(◡𝐹‘𝑥)))))
40	35, 39	rspc2v 3293	. . . . . . . . . . . 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)))))
41	40	impancom 455	. . . . . . . . . . 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)))))
42	13, 31, 41	syl2anc 691	. . . . . . . . . 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)))))
43	42	imp 444	. . . . . . . . 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))))
44	28, 43	syldan 486	. . . . . . . 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))))
45	44	breq1d 4593	. . . . . . 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))
46	30, 45	bitrd 267	. . . . . 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))
47		f1of1 6049	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌)
48	5, 47	syl 17	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → 𝐹:𝑋–1-1→𝑌)
49	48	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋–1-1→𝑌)
50		blssm 22033	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
51	20, 13, 16, 50	syl3anc 1318	. . . . . . . 8 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
52	51	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
53		f1elima 6421	. . . . . . 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)))
54	49, 28, 52, 53	syl3anc 1318	. . . . . 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)))
55	12	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑁 ∈ (∞Met‘𝑌))
56	15	adantr 480	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘𝑃) ∈ 𝑌)
57		f1ocnvfv2 6433	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
58	5, 57	sylan 487	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥)
59		simpr 476	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ 𝑌)
60	58, 59	eqeltrd 2688	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) ∈ 𝑌)
61		elbl2 22005	. . . . . . 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))
62	55, 22, 56, 60, 61	syl22anc 1319	. . . . . 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))
63	46, 54, 62	3bitr4d 299	. . . . 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)))
64	58	eleq1d 2672	. . . . 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))))
65	58	eleq1d 2672	. . . . 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)))
66	63, 64, 65	3bitr3d 297	. . . 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)))
67	66	ex 449	. . 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))))
68	11, 19, 67	pm5.21ndd 368	. 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)))
69	68	eqrdv 2608	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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   class class class wbr 4583  ^◡ccnv 5037  ran crn 5039   “ cima 5041  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  ℝ^*cxr 9952   < clt 9953  ∞Metcxmt 19552  ballcbl 19554   Ismty cismty 32767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746  df-xr 9957  df-psmet 19559  df-xmet 19560  df-bl 19562  df-ismty 32768

This theorem is referenced by:  ismtyhmeolem  32773  ismtybndlem  32775