Theorem ispautN 40208

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 40207	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑀 = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
4	3	eleq2d 2817	. 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 6763	. . . . 5 ⊢ (𝐹:𝑆–1-1-onto→𝑆 → 𝐹:𝑆⟶𝑆)
6	1	fvexi 6836	. . . . 5 ⊢ 𝑆 ∈ V
7		fex 7160	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑆 ∧ 𝑆 ∈ V) → 𝐹 ∈ V)
8	5, 6, 7	sylancl 586	. . . 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 6751	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝑆–1-1-onto→𝑆 ↔ 𝐹:𝑆–1-1-onto→𝑆))
11		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
12		fveq1 6821	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
13	11, 12	sseq12d 3963	. . . . . 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 3196	. . . 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 632	. . 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 3637	. 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 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  Vcvv 3436   ⊆ wss 3897  ⟶wf 6477  –1-1-onto→wf1o 6480  ‘cfv 6481  PSubSpcpsubsp 39605  PAutcpautN 40096

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752  df-pautN 40100

This theorem is referenced by: (None)