Theorem pautsetN 40099

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 3471	. 2 ⊢ (𝐾 ∈ 𝐵 → 𝐾 ∈ V)
2		pautset.m	. . 3 ⊢ 𝑀 = (PAut‘𝐾)
3		fveq2 6861	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = (PSubSp‘𝐾))
4		pautset.s	. . . . . . . . 9 ⊢ 𝑆 = (PSubSp‘𝐾)
5	3, 4	eqtr4di 2783	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (PSubSp‘𝑘) = 𝑆)
6	5	f1oeq2d 6799	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ↔ 𝑓:𝑆–1-1-onto→(PSubSp‘𝑘)))
7		f1oeq3 6793	. . . . . . . 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 3301	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (PSubSp‘𝑘)(𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)) ↔ ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦))))
11	5, 10	raleqbidv 3321	. . . . . 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 2796	. . . 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 39992	. . . 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 6875	. . . . . . . 8 ⊢ 𝑆 ∈ V
16	15, 15	mapval 8814	. . . . . . 7 ⊢ (𝑆 ↑m 𝑆) = {𝑓 ∣ 𝑓:𝑆⟶𝑆}
17		ovex 7423	. . . . . . 7 ⊢ (𝑆 ↑m 𝑆) ∈ V
18	16, 17	eqeltrri 2826	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝑆⟶𝑆} ∈ V
19		f1of 6803	. . . . . . 7 ⊢ (𝑓:𝑆–1-1-onto→𝑆 → 𝑓:𝑆⟶𝑆)
20	19	ss2abi 4033	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝑆–1-1-onto→𝑆} ⊆ {𝑓 ∣ 𝑓:𝑆⟶𝑆}
21	18, 20	ssexi 5280	. . . . 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 4033	. . . . 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 5280	. . . 4 ⊢ {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))} ∈ V
25	13, 14, 24	fvmpt 6971	. . 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 2777	. 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 2109  {cab 2708  ∀wral 3045  Vcvv 3450   ⊆ wss 3917  ⟶wf 6510  –1-1-onto→wf1o 6513  ‘cfv 6514  (class class class)co 7390   ↑_m cmap 8802  PSubSpcpsubsp 39497  PAutcpautN 39988

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-map 8804  df-pautN 39992

This theorem is referenced by:  ispautN  40100