Theorem ispautN 40479

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 40478	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑀 = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
4	3	eleq2d 2823	. 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 6782	. . . . 5 ⊢ (𝐹:𝑆–1-1-onto→𝑆 → 𝐹:𝑆⟶𝑆)
6	1	fvexi 6856	. . . . 5 ⊢ 𝑆 ∈ V
7		fex 7182	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑆 ∧ 𝑆 ∈ V) → 𝐹 ∈ V)
8	5, 6, 7	sylancl 587	. . . 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 6770	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝑆–1-1-onto→𝑆 ↔ 𝐹:𝑆–1-1-onto→𝑆))
11		fveq1 6841	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
12		fveq1 6841	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
13	11, 12	sseq12d 3969	. . . . . 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 3202	. . . 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 633	. . 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 3643	. 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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  Vcvv 3442   ⊆ wss 3903  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  PSubSpcpsubsp 39876  PAutcpautN 40367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-map 8777  df-pautN 40371

This theorem is referenced by: (None)