Theorem pautsetN 40100

Description: The set of projective automorphisms. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)

Hypotheses

Ref	Expression
pautset.s	⊢ 𝑆 = (PSubSp‘𝐾)
pautset.m	⊢ 𝑀 = (PAut‘𝐾)

Assertion

Ref	Expression
pautsetN	⊢ (𝐾 ∈ 𝐵 → 𝑀 = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})

Distinct variable groups:   𝑥,𝑓,𝑦   𝑓,𝐾,𝑥   𝑆,𝑓,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓)   𝐾(𝑦)   𝑀(𝑥,𝑦,𝑓)

Proof of Theorem pautsetN

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3501	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		pautset.m	. . 3 ⊢ 𝑀 = (PAut‘𝐾)
3		fveq2 6906	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
4		pautset.s	. . . . . . . . 9 ⊢ 𝑆 = (PSubSp‘𝐾)
5	3, 4	eqtr4di 2795	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
6	5	f1oeq2d 6844	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆–1-1-onto→(PSubSp‘𝑘)))
7		f1oeq3 6838	. . . . . . . 8 ⊢ ((PSubSp‘𝑘) = 𝑆 → (𝑓:𝑆–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆–1-1-onto→𝑆))
8	5, 7	syl 17	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:𝑆–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆–1-1-onto→𝑆))
9	6, 8	bitrd 279	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆–1-1-onto→𝑆))
10	5	raleqdv 3326	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦))))
11	5, 10	raleqbidv 3346	. . . . . 6 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦))))
12	9, 11	anbi12d 632	. . . . 5 ⊢ (𝑘 = 𝐾 → ((𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦))) ↔ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))))
13	12	abbidv 2808	. . . 4 ⊢ (𝑘 = 𝐾 → {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))} = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
14		df-pautN 39993	. . . 4 ⊢ PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
15	4	fvexi 6920	. . . . . . . 8 ⊢ 𝑆 ∈ V
16	15, 15	mapval 8878	. . . . . . 7 ⊢ (𝑆 ↑m 𝑆) = {𝑓 ∣ 𝑓:𝑆⟶𝑆}
17		ovex 7464	. . . . . . 7 ⊢ (𝑆 ↑m 𝑆) ∈ V
18	16, 17	eqeltrri 2838	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝑆⟶𝑆} ∈ V
19		f1of 6848	. . . . . . 7 ⊢ (𝑓:𝑆–1-1-onto→𝑆 → 𝑓:𝑆⟶𝑆)
20	19	ss2abi 4067	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝑆–1-1-onto→𝑆} ⊆ {𝑓 ∣ 𝑓:𝑆⟶𝑆}
21	18, 20	ssexi 5322	. . . . 5 ⊢ {𝑓 ∣ 𝑓:𝑆–1-1-onto→𝑆} ∈ V
22		simpl 482	. . . . . 6 ⊢ ((𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦))) → 𝑓:𝑆–1-1-onto→𝑆)
23	22	ss2abi 4067	. . . . 5 ⊢ {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))} ⊆ {𝑓 ∣ 𝑓:𝑆–1-1-onto→𝑆}
24	21, 23	ssexi 5322	. . . 4 ⊢ {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))} ∈ V
25	13, 14, 24	fvmpt 7016	. . 3 ⊢ (𝐾 ∈ V → (PAut‘𝐾) = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
26	2, 25	eqtrid 2789	. 2 ⊢ (𝐾 ∈ V → 𝑀 = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
27	1, 26	syl 17	1 ⊢ (𝐾 ∈ 𝐵 → 𝑀 = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {cab 2714  ∀wral 3061  Vcvv 3480   ⊆ wss 3951  ⟶wf 6557  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  PSubSpcpsubsp 39498  PAutcpautN 39989

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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-pautN 39993

This theorem is referenced by:  ispautN  40101