ismtyima - Mathbox for Jeff Madsen

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Theorem ismtyima 38831

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 6197	. . . . 5 ⊢ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2		isismty 38829	. . . . . . . . . 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 6825	. . . . . . 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 6719	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) → ran 𝐹 ⊆ 𝑌)
9	1, 8	sstrid 3919	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
10	9	sseld 3907	. . 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 7082	. . . . . 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 24743	. . . . 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 3907	. . 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 6838	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
24		f1of 6825	. . . . . . . . . 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 7082	. . . . . . . . 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 24715	. . . . . . . 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 7428	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
32		fveq2 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑃 → (𝐹‘𝑥) = (𝐹‘𝑃))
33	32	oveq1d 7436	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)))
34	31, 33	eqeq12d 2776	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦))))
35		oveq2 7429	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (^◡𝐹‘𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(^◡𝐹‘𝑥)))
36		fveq2 6886	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (^◡𝐹‘𝑥) → (𝐹‘𝑦) = (𝐹‘(^◡𝐹‘𝑥)))
37	36	oveq2d 7437	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (^◡𝐹‘𝑥) → ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘(^◡𝐹‘𝑥))))
38	35, 37	eqeq12d 2776	. . . . . . . . . . . . 13 ⊢ (𝑦 = (^◡𝐹‘𝑥) → ((𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀(^◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(^◡𝐹‘𝑥)))))
39	34, 38	rspc2v 3586	. . . . . . . . . . . 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 5112	. . . . . . 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 6824	. . . . . . . . 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 24743	. . . . . . . . 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 7268	. . . . . . 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 7286	. . . . . . . . 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 2860	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) ∈ 𝑌)
60		elbl2 24715	. . . . . . 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 2845	. . . . 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 2845	. . . . 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 2758	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))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ∀wral 3076 ⊆ wss 3895 class class class wbr 5102 ^◡ccnv 5649 ran crn 5651 “ cima 5653 ⟶wf 6535 –1-1→wf1 6536 –1-1-onto→wf1o 6538 ‘cfv 6539 (class class class)co 7421 ℝ^*cxr 11346 < clt 11347 ∞Metcxmet 21669 ballcbl 21671 Ismty cismty 38826

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7752 ax-cnex 11260 ax-resscn 11261

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-ov 7424 df-oprab 7425 df-mpo 7426 df-1st 8002 df-2nd 8003 df-map 8853 df-xr 11351 df-psmet 21676 df-xmet 21677 df-bl 21679 df-ismty 38827

This theorem is used by: ismtyhmeolem 38832 ismtybndlem 38834

W3C validator