Theorem ispautN 35385

Description: The predictate "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 35384	. . 3 ⊢ (𝐾 ∈ 𝐵 → 𝑀 = {𝑓 ∣ (𝑓:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)))})
4	3	eleq2d 2687	. 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 6137	. . . . 5 ⊢ (𝐹:𝑆–1-1-onto→𝑆 → 𝐹:𝑆⟶𝑆)
6		fvex 6201	. . . . . 6 ⊢ (PSubSp‘𝐾) ∈ V
7	1, 6	eqeltri 2697	. . . . 5 ⊢ 𝑆 ∈ V
8		fex 6490	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑆 ∧ 𝑆 ∈ V) → 𝐹 ∈ V)
9	5, 7, 8	sylancl 694	. . . 4 ⊢ (𝐹:𝑆–1-1-onto→𝑆 → 𝐹 ∈ V)
10	9	adantr 481	. . 3 ⊢ ((𝐹:𝑆–1-1-onto→𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦))) → 𝐹 ∈ V)
11		f1oeq1 6127	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑓:𝑆–1-1-onto→𝑆 ↔ 𝐹:𝑆–1-1-onto→𝑆))
12		fveq1 6190	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13		fveq1 6190	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
14	12, 13	sseq12d 3634	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑓‘𝑥) ⊆ (𝑓‘𝑦) ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦)))
15	14	bibi2d 332	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)) ↔ (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦))))
16	15	2ralbidv 2989	. . . 4 ⊢ (𝑓 = 𝐹 → (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝑓‘𝑥) ⊆ (𝑓‘𝑦)) ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 ⊆ 𝑦 ↔ (𝐹‘𝑥) ⊆ (𝐹‘𝑦))))
17	11, 16	anbi12d 747	. . 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)))))
18	10, 17	elab3 3358	. 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))))
19	4, 18	syl6bb 276	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  Vcvv 3200   ⊆ wss 3574  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  PSubSpcpsubsp 34782  PAutcpautN 35273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-pautN 35277

This theorem is referenced by: (None)