Theorem pmtrfv 19400

Description: General value of mapping a point under a transposition. (Contributed by Stefan O'Rear, 16-Aug-2015.)

Hypothesis

Ref	Expression
pmtrfval.t	⊢ 𝑇 = (pmTrsp‘𝐷)

Assertion

Ref	Expression
pmtrfv	⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))

Proof of Theorem pmtrfv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pmtrfval.t	. . . . 5 ⊢ 𝑇 = (pmTrsp‘𝐷)
2	1	pmtrval 19399	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
3	2	fveq1d 6893	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → ((𝑇‘𝑃)‘𝑍) = ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
4	3	adantr 480	. 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
5		eqid 2727	. . 3 ⊢ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
6		eleq1 2816	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ∈ 𝑃 ↔ 𝑍 ∈ 𝑃))
7		sneq 4634	. . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍})
8	7	difeq2d 4118	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
9	8	unieqd 4916	. . . 4 ⊢ (𝑧 = 𝑍 → ∪ (𝑃 ∖ {𝑧}) = ∪ (𝑃 ∖ {𝑍}))
10		id 22	. . . 4 ⊢ (𝑧 = 𝑍 → 𝑧 = 𝑍)
11	6, 9, 10	ifbieq12d 4552	. . 3 ⊢ (𝑧 = 𝑍 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))
12		simpr 484	. . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑍 ∈ 𝐷)
13		simpl3 1191	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑃 ≈ 2o)
14		relen 8962	. . . . . 6 ⊢ Rel ≈
15	14	brrelex1i 5728	. . . . 5 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ V)
16		difexg 5323	. . . . 5 ⊢ (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
17		uniexg 7739	. . . . 5 ⊢ ((𝑃 ∖ {𝑍}) ∈ V → ∪ (𝑃 ∖ {𝑍}) ∈ V)
18	13, 15, 16, 17	4syl 19	. . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ∪ (𝑃 ∖ {𝑍}) ∈ V)
19		ifexg 4573	. . . 4 ⊢ ((∪ (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
20	18, 19	sylancom 587	. . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
21	5, 11, 12, 20	fvmptd3 7022	. 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))
22	4, 21	eqtrd 2767	1 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  Vcvv 3469   ∖ cdif 3941   ⊆ wss 3944  ifcif 4524  {csn 4624  ∪ cuni 4903   class class class wbr 5142   ↦ cmpt 5225  ‘cfv 6542  2_oc2o 8474   ≈ cen 8954  pmTrspcpmtr 19389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-en 8958  df-pmtr 19390

This theorem is referenced by:  pmtrprfv  19401  pmtrprfv3  19402  pmtrmvd  19404  pmtrffv  19407