ismtyima - Mathbox for Jeff Madsen

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Theorem ismtyima 37184

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 6064	. . . . 5 ⊢ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2		isismty 37182	. . . . . . . . . 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 1465	. . . . . . . . 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 6827	. . . . . . 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 6719	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) → ran 𝐹 ⊆ 𝑌)
9	1, 8	sstrid 3988	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
10	9	sseld 3976	. . 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 1189	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) → 𝑁 ∈ (∞Met‘𝑌))
12		simprl 768	. . . . . 6 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) → 𝑃 ∈ 𝑋)
13		ffvelcdm 7077	. . . . . 6 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑃 ∈ 𝑋) → (𝐹‘𝑃) ∈ 𝑌)
14	7, 12, 13	syl2anc 583	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) → (𝐹‘𝑃) ∈ 𝑌)
15		simprr 770	. . . . 5 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^)) → 𝑅 ∈ ℝ^)
16		blssm 24279	. . . . 5 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹‘𝑃) ∈ 𝑌 ∧ 𝑅 ∈ ℝ^*) → ((𝐹‘𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
17	11, 14, 15, 16	syl3anc 1368	. . . 4 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) → ((𝐹‘𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
18	17	sseld 3976	. . 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 1188	. . . . . . . . 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 775	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^)) ∧ 𝑥 ∈ 𝑌) → 𝑅 ∈ ℝ^)
22		simplrl 774	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) ∧ 𝑥 ∈ 𝑌) → 𝑃 ∈ 𝑋)
23		f1ocnv 6839	. . . . . . . . . 10 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
24		f1of 6827	. . . . . . . . . 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 7077	. . . . . . . . 9 ⊢ ((^◡𝐹:𝑌⟶𝑋 ∧ 𝑥 ∈ 𝑌) → (^◡𝐹‘𝑥) ∈ 𝑋)
27	25, 26	sylan 579	. . . . . . . 8 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) ∧ 𝑥 ∈ 𝑌) → (^◡𝐹‘𝑥) ∈ 𝑋)
28		elbl2 24251	. . . . . . . 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 836	. . . . . . 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 7412	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
32		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑃 → (𝐹‘𝑥) = (𝐹‘𝑃))
33	32	oveq1d 7420	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑃 → ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)))
34	31, 33	eqeq12d 2742	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦))))
35		oveq2 7413	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (^◡𝐹‘𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(^◡𝐹‘𝑥)))
36		fveq2 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (^◡𝐹‘𝑥) → (𝐹‘𝑦) = (𝐹‘(^◡𝐹‘𝑥)))
37	36	oveq2d 7421	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (^◡𝐹‘𝑥) → ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) = ((𝐹‘𝑃)𝑁(𝐹‘(^◡𝐹‘𝑥))))
38	35, 37	eqeq12d 2742	. . . . . . . . . . . . 13 ⊢ (𝑦 = (^◡𝐹‘𝑥) → ((𝑃𝑀𝑦) = ((𝐹‘𝑃)𝑁(𝐹‘𝑦)) ↔ (𝑃𝑀(^◡𝐹‘𝑥)) = ((𝐹‘𝑃)𝑁(𝐹‘(^◡𝐹‘𝑥)))))
39	34, 38	rspc2v 3617	. . . . . . . . . . . 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 583	. . . . . . . . . 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 590	. . . . . . . 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 5151	. . . . . . 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 6826	. . . . . . . . 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 24279	. . . . . . . . 9 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
50	19, 12, 15, 49	syl3anc 1368	. . . . . . . 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 7258	. . . . . . 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 1368	. . . . . 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 7271	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
57	5, 56	sylan 579	. . . . . . . 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 2827	. . . . . . 7 ⊢ ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ^*)) ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) ∈ 𝑌)
60		elbl2 24251	. . . . . . 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 836	. . . . . 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 2812	. . . . 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 2812	. . . . 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 2724	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

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3055 ⊆ wss 3943 class class class wbr 5141 ^◡ccnv 5668 ran crn 5670 “ cima 5672 ⟶wf 6533 –1-1→wf1 6534 –1-1-onto→wf1o 6536 ‘cfv 6537 (class class class)co 7405 ℝ^*cxr 11251 < clt 11252 ∞Metcxmet 21225 ballcbl 21227 Ismty cismty 37179

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-map 8824 df-xr 11256 df-psmet 21232 df-xmet 21233 df-bl 21235 df-ismty 37180

This theorem is referenced by: ismtyhmeolem 37185 ismtybndlem 37187

W3C validator