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Theorem ismtyima 33915
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)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) = ((𝐹𝑃)(ball‘𝑁)𝑅))

Proof of Theorem ismtyima
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imassrn 5635 . . . . 5 (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ ran 𝐹
2 isismty 33913 . . . . . . . . . 10 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))))
32biimp3a 1581 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) → (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))))
43adantr 472 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹:𝑋1-1-onto𝑌 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))))
54simpld 477 . . . . . . 7 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋1-1-onto𝑌)
6 f1of 6298 . . . . . . 7 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋𝑌)
75, 6syl 17 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋𝑌)
8 frn 6214 . . . . . 6 (𝐹:𝑋𝑌 → ran 𝐹𝑌)
97, 8syl 17 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ran 𝐹𝑌)
101, 9syl5ss 3755 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ⊆ 𝑌)
1110sseld 3743 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) → 𝑥𝑌))
12 simpl2 1230 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑁 ∈ (∞Met‘𝑌))
13 simprl 811 . . . . . 6 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑃𝑋)
14 ffvelrn 6520 . . . . . 6 ((𝐹:𝑋𝑌𝑃𝑋) → (𝐹𝑃) ∈ 𝑌)
157, 13, 14syl2anc 696 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹𝑃) ∈ 𝑌)
16 simprr 813 . . . . 5 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑅 ∈ ℝ*)
17 blssm 22424 . . . . 5 ((𝑁 ∈ (∞Met‘𝑌) ∧ (𝐹𝑃) ∈ 𝑌𝑅 ∈ ℝ*) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1812, 15, 16, 17syl3anc 1477 . . . 4 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑃)(ball‘𝑁)𝑅) ⊆ 𝑌)
1918sseld 3743 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) → 𝑥𝑌))
20 simpl1 1228 . . . . . . . . 9 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝑀 ∈ (∞Met‘𝑋))
2120adantr 472 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑀 ∈ (∞Met‘𝑋))
22 simplrr 820 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑅 ∈ ℝ*)
23 simplrl 819 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑃𝑋)
24 f1ocnv 6310 . . . . . . . . . 10 (𝐹:𝑋1-1-onto𝑌𝐹:𝑌1-1-onto𝑋)
25 f1of 6298 . . . . . . . . . 10 (𝐹:𝑌1-1-onto𝑋𝐹:𝑌𝑋)
265, 24, 253syl 18 . . . . . . . . 9 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑌𝑋)
27 ffvelrn 6520 . . . . . . . . 9 ((𝐹:𝑌𝑋𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
2826, 27sylan 489 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹𝑥) ∈ 𝑋)
29 elbl2 22396 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑅 ∈ ℝ*) ∧ (𝑃𝑋 ∧ (𝐹𝑥) ∈ 𝑋)) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(𝐹𝑥)) < 𝑅))
3021, 22, 23, 28, 29syl22anc 1478 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ (𝑃𝑀(𝐹𝑥)) < 𝑅))
314simprd 482 . . . . . . . . . . 11 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)))
32 oveq1 6820 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → (𝑥𝑀𝑦) = (𝑃𝑀𝑦))
33 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑥 = 𝑃 → (𝐹𝑥) = (𝐹𝑃))
3433oveq1d 6828 . . . . . . . . . . . . . 14 (𝑥 = 𝑃 → ((𝐹𝑥)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹𝑦)))
3532, 34eqeq12d 2775 . . . . . . . . . . . . 13 (𝑥 = 𝑃 → ((𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) ↔ (𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦))))
36 oveq2 6821 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → (𝑃𝑀𝑦) = (𝑃𝑀(𝐹𝑥)))
37 fveq2 6352 . . . . . . . . . . . . . . 15 (𝑦 = (𝐹𝑥) → (𝐹𝑦) = (𝐹‘(𝐹𝑥)))
3837oveq2d 6829 . . . . . . . . . . . . . 14 (𝑦 = (𝐹𝑥) → ((𝐹𝑃)𝑁(𝐹𝑦)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
3936, 38eqeq12d 2775 . . . . . . . . . . . . 13 (𝑦 = (𝐹𝑥) → ((𝑃𝑀𝑦) = ((𝐹𝑃)𝑁(𝐹𝑦)) ↔ (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4035, 39rspc2v 3461 . . . . . . . . . . . 12 ((𝑃𝑋 ∧ (𝐹𝑥) ∈ 𝑋) → (∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦)) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4140impancom 455 . . . . . . . . . . 11 ((𝑃𝑋 ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝑀𝑦) = ((𝐹𝑥)𝑁(𝐹𝑦))) → ((𝐹𝑥) ∈ 𝑋 → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4213, 31, 41syl2anc 696 . . . . . . . . . 10 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → ((𝐹𝑥) ∈ 𝑋 → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥)))))
4342imp 444 . . . . . . . . 9 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ (𝐹𝑥) ∈ 𝑋) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
4428, 43syldan 488 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑃𝑀(𝐹𝑥)) = ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))))
4544breq1d 4814 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝑃𝑀(𝐹𝑥)) < 𝑅 ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
4630, 45bitrd 268 . . . . . 6 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅) ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
47 f1of1 6297 . . . . . . . . 9 (𝐹:𝑋1-1-onto𝑌𝐹:𝑋1-1𝑌)
485, 47syl 17 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → 𝐹:𝑋1-1𝑌)
4948adantr 472 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝐹:𝑋1-1𝑌)
50 blssm 22424 . . . . . . . . 9 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ*) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5120, 13, 16, 50syl3anc 1477 . . . . . . . 8 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
5251adantr 472 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋)
53 f1elima 6683 . . . . . . 7 ((𝐹:𝑋1-1𝑌 ∧ (𝐹𝑥) ∈ 𝑋 ∧ (𝑃(ball‘𝑀)𝑅) ⊆ 𝑋) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅)))
5449, 28, 52, 53syl3anc 1477 . . . . . 6 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (𝐹𝑥) ∈ (𝑃(ball‘𝑀)𝑅)))
5512adantr 472 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑁 ∈ (∞Met‘𝑌))
5615adantr 472 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹𝑃) ∈ 𝑌)
57 f1ocnvfv2 6696 . . . . . . . . 9 ((𝐹:𝑋1-1-onto𝑌𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
585, 57sylan 489 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) = 𝑥)
59 simpr 479 . . . . . . . 8 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → 𝑥𝑌)
6058, 59eqeltrd 2839 . . . . . . 7 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝐹‘(𝐹𝑥)) ∈ 𝑌)
61 elbl2 22396 . . . . . . 7 (((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑅 ∈ ℝ*) ∧ ((𝐹𝑃) ∈ 𝑌 ∧ (𝐹‘(𝐹𝑥)) ∈ 𝑌)) → ((𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
6255, 22, 56, 60, 61syl22anc 1478 . . . . . 6 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) ↔ ((𝐹𝑃)𝑁(𝐹‘(𝐹𝑥))) < 𝑅))
6346, 54, 623bitr4d 300 . . . . 5 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ (𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6458eleq1d 2824 . . . . 5 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅))))
6558eleq1d 2824 . . . . 5 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → ((𝐹‘(𝐹𝑥)) ∈ ((𝐹𝑃)(ball‘𝑁)𝑅) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6663, 64, 653bitr3d 298 . . . 4 ((((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) ∧ 𝑥𝑌) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6766ex 449 . . 3 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥𝑌 → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅))))
6811, 19, 67pm5.21ndd 368 . 2 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝑥 ∈ (𝐹 “ (𝑃(ball‘𝑀)𝑅)) ↔ 𝑥 ∈ ((𝐹𝑃)(ball‘𝑁)𝑅)))
6968eqrdv 2758 1 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌) ∧ 𝐹 ∈ (𝑀 Ismty 𝑁)) ∧ (𝑃𝑋𝑅 ∈ ℝ*)) → (𝐹 “ (𝑃(ball‘𝑀)𝑅)) = ((𝐹𝑃)(ball‘𝑁)𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  wss 3715   class class class wbr 4804  ccnv 5265  ran crn 5267  cima 5269  wf 6045  1-1wf1 6046  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  *cxr 10265   < clt 10266  ∞Metcxmt 19933  ballcbl 19935   Ismty cismty 33910
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-map 8025  df-xr 10270  df-psmet 19940  df-xmet 19941  df-bl 19943  df-ismty 33911
This theorem is referenced by:  ismtyhmeolem  33916  ismtybndlem  33918
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