Theorem ispautN 39482

Description: The predicate "is a projective automorphism". (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
ispautN	⊢ (𝐾 ∈ 𝐵 → (𝐹 ∈ 𝑀 ↔ (𝐹:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))))

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

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

Proof of Theorem ispautN

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pautset.s	. . . 4 ⊢ 𝑆 = (PSubSp‘𝐾)
2		pautset.m	. . . 4 ⊢ 𝑀 = (PAut‘𝐾)
3	1, 2	pautsetN 39481	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑀 = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
4	3	eleq2d 2813	. 2 ⊢ (𝐾 ∈ 𝐵 → (𝐹 ∈ 𝑀 ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))}))
5		f1of 6826	. . . . 5 ⊢ (𝐹:𝑆–1-1-onto→𝑆 → 𝐹:𝑆⟶𝑆)
6	1	fvexi 6898	. . . . 5 ⊢ 𝑆 ∈ V
7		fex 7222	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑆 ∧ 𝑆 ∈ V) → 𝐹 ∈ V)
8	5, 6, 7	sylancl 585	. . . 4 ⊢ (𝐹:𝑆–1-1-onto→𝑆 → 𝐹 ∈ V)
9	8	adantr 480	. . 3 ⊢ ((𝐹:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦))) → 𝐹 ∈ V)
10		f1oeq1 6814	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝑆–1-1-onto→𝑆 ↔ 𝐹:𝑆–1-1-onto→𝑆))
11		fveq1 6883	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
12		fveq1 6883	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
13	11, 12	sseq12d 4010	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ⊆ (𝑓‘𝑦) ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))
14	13	bibi2d 342	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)) ↔ (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦))))
15	14	2ralbidv 3212	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦))))
16	10, 15	anbi12d 630	. . 3 ⊢ (𝑓 = 𝐹 → ((𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦))) ↔ (𝐹:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))))
17	9, 16	elab3 3671	. 2 ⊢ (𝐹 ∈ {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))} ↔ (𝐹:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦))))
18	4, 17	bitrdi 287	1 ⊢ (𝐾 ∈ 𝐵 → (𝐹 ∈ 𝑀 ↔ (𝐹:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {cab 2703  ∀wral 3055  Vcvv 3468   ⊆ wss 3943  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  PSubSpcpsubsp 38879  PAutcpautN 39370

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-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

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-reu 3371  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 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821  df-pautN 39374

This theorem is referenced by: (None)