Theorem pmtrfv 19445

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 19444	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
3	2	fveq1d 6894	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → ((𝑇‘𝑃)‘𝑍) = ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
4	3	adantr 479	. 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
5		eqid 2726	. . 3 ⊢ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
6		eleq1 2814	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ∈ 𝑃 ↔ 𝑍 ∈ 𝑃))
7		sneq 4635	. . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍})
8	7	difeq2d 4120	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
9	8	unieqd 4920	. . . 4 ⊢ (𝑧 = 𝑍 → ∪ (𝑃 ∖ {𝑧}) = ∪ (𝑃 ∖ {𝑍}))
10		id 22	. . . 4 ⊢ (𝑧 = 𝑍 → 𝑧 = 𝑍)
11	6, 9, 10	ifbieq12d 4553	. . 3 ⊢ (𝑧 = 𝑍 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))
12		simpr 483	. . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑍 ∈ 𝐷)
13		simpl3 1190	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑃 ≈ 2o)
14		relen 8970	. . . . . 6 ⊢ Rel ≈
15	14	brrelex1i 5730	. . . . 5 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ V)
16		difexg 5326	. . . . 5 ⊢ (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
17		uniexg 7742	. . . . 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 4574	. . . 4 ⊢ ((∪ (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
20	18, 19	sylancom 586	. . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
21	5, 11, 12, 20	fvmptd3 7023	. 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 2766	1 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  Vcvv 3464   ∖ cdif 3945   ⊆ wss 3948  ifcif 4525  {csn 4625  ∪ cuni 4907   class class class wbr 5145   ↦ cmpt 5228  ‘cfv 6545  2_oc2o 8481   ≈ cen 8962  pmTrspcpmtr 19434

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 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3466  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4908  df-iun 4997  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6497  df-fun 6547  df-fn 6548  df-f 6549  df-f1 6550  df-fo 6551  df-f1o 6552  df-fv 6553  df-en 8966  df-pmtr 19435

This theorem is referenced by:  pmtrprfv  19446  pmtrprfv3  19447  pmtrmvd  19449  pmtrffv  19452