Theorem isismty 37825

Description: The condition "is an isometry". (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
isismty	⊢ ((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)))))

Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑁,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝐹,𝑦

Proof of Theorem isismty

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismtyval 37824	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝑀 Ismty 𝑁) = {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))})
2	1	eleq2d 2820	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ (𝑀 Ismty 𝑁) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))}))
3		f1of 6818	. . . . . . 7 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋⟶𝑌)
4	3	adantr 480	. . . . . 6 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → 𝐹:𝑋⟶𝑌)
5		elfvdm 6913	. . . . . 6 ⊢ (𝑀 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
6		elfvdm 6913	. . . . . 6 ⊢ (𝑁 ∈ (∞Met‘𝑌) → 𝑌 ∈ dom ∞Met)
7		fex2 7932	. . . . . 6 ⊢ ((𝐹:𝑋⟶𝑌 ∧ 𝑋 ∈ dom ∞Met ∧ 𝑌 ∈ dom ∞Met) → 𝐹 ∈ V)
8	4, 5, 6, 7	syl3an 1160	. . . . 5 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) ∧ 𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝐹 ∈ V)
9	8	3expib 1122	. . . 4 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → 𝐹 ∈ V))
10	9	com12 32	. . 3 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → ((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → 𝐹 ∈ V))
11		f1oeq1 6806	. . . . 5 ⊢ (𝑓 = 𝐹 → (𝑓:𝑋–1-1-onto→𝑌 ↔ 𝐹:𝑋–1-1-onto→𝑌))
12		fveq1 6875	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13		fveq1 6875	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
14	12, 13	oveq12d 7423	. . . . . . 7 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥)𝑁(𝑓‘𝑦)) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))
15	14	eqeq2d 2746	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)) ↔ (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))
16	15	2ralbidv 3205	. . . . 5 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))))
17	11, 16	anbi12d 632	. . . 4 ⊢ (𝑓 = 𝐹 → ((𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦))) ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))))
18	17	elab3g 3664	. . 3 ⊢ (((𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦))) → 𝐹 ∈ V) → (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))} ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))))
19	10, 18	syl 17	. 2 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑁 ∈ (∞Met‘𝑌)) → (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝑓‘𝑥)𝑁(𝑓‘𝑦)))} ↔ (𝐹:𝑋–1-1-onto→𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝑀𝑦) = ((𝐹‘𝑥)𝑁(𝐹‘𝑦)))))
20	2, 19	bitrd 279	1 ⊢ ((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)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2713  ∀wral 3051  Vcvv 3459  dom cdm 5654  ⟶wf 6527  –1-1-onto→wf1o 6530  ‘cfv 6531  (class class class)co 7405  ∞Metcxmet 21300   Ismty cismty 37822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8842  df-xr 11273  df-xmet 21308  df-ismty 37823

This theorem is referenced by:  ismtycnv  37826  ismtyima  37827  ismtyhmeolem  37828  ismtybndlem  37830  ismtyres  37832  ismrer1  37862  reheibor  37863