Theorem pmtrfv 19523

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 19522	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
3	2	fveq1d 6885	. . 3 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → ((𝑇‘𝑃)‘𝑍) = ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
4	3	adantr 485	. 2 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = ((𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))‘𝑍))
5		eqid 2763	. . 3 ⊢ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
6		eleq1 2851	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ∈ 𝑃 ↔ 𝑍 ∈ 𝑃))
7		sneq 4600	. . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍})
8	7	difeq2d 4082	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
9	8	unieqd 4886	. . . 4 ⊢ (𝑧 = 𝑍 → ∪ (𝑃 ∖ {𝑧}) = ∪ (𝑃 ∖ {𝑍}))
10		id 23	. . . 4 ⊢ (𝑧 = 𝑍 → 𝑧 = 𝑍)
11	6, 9, 10	ifbieq12d 4517	. . 3 ⊢ (𝑧 = 𝑍 → if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))
12		simpr 489	. . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑍 ∈ 𝐷)
13		simpl3 1212	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑃 ≈ 2o)
14		relen 8949	. . . . . 6 ⊢ Rel ≈
15	14	brrelex1i 5719	. . . . 5 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ V)
16		difexg 5301	. . . . 5 ⊢ (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
17		uniexg 7740	. . . . 5 ⊢ ((𝑃 ∖ {𝑍}) ∈ V → ∪ (𝑃 ∖ {𝑍}) ∈ V)
18	13, 15, 16, 17	4syl 20	. . . 4 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ∪ (𝑃 ∖ {𝑍}) ∈ V)
19		ifexg 4538	. . . 4 ⊢ ((∪ (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
20	18, 19	sylancom 599	. . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
21	5, 11, 12, 20	fvmptd3 7015	. 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 2798	1 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → ((𝑇‘𝑃)‘𝑍) = if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  Vcvv 3455   ∖ cdif 3903   ⊆ wss 3906  ifcif 4488  {csn 4590  ∪ cuni 4873   class class class wbr 5110   ↦ cmpt 5193  ‘cfv 6538  2_oc2o 8448   ≈ cen 8941  pmTrspcpmtr 19512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-en 8945  df-pmtr 19513

This theorem is referenced by:  pmtrprfv  19524  pmtrprfv3  19525  pmtrmvd  19527  pmtrffv  19530