Theorem pmtrfv 19393

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 19392	. . . 4 ⊢ ((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) → (𝑇‘𝑃) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)))
3	2	fveq1d 6844	. . 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 2737	. . 3 ⊢ (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧)) = (𝑧 ∈ 𝐷 ↦ if(𝑧 ∈ 𝑃, ∪ (𝑃 ∖ {𝑧}), 𝑧))
6		eleq1 2825	. . . 4 ⊢ (𝑧 = 𝑍 → (𝑧 ∈ 𝑃 ↔ 𝑍 ∈ 𝑃))
7		sneq 4592	. . . . . 6 ⊢ (𝑧 = 𝑍 → {𝑧} = {𝑍})
8	7	difeq2d 4080	. . . . 5 ⊢ (𝑧 = 𝑍 → (𝑃 ∖ {𝑧}) = (𝑃 ∖ {𝑍}))
9	8	unieqd 4878	. . . 4 ⊢ (𝑧 = 𝑍 → ∪ (𝑃 ∖ {𝑧}) = ∪ (𝑃 ∖ {𝑍}))
10		id 22	. . . 4 ⊢ (𝑧 = 𝑍 → 𝑧 = 𝑍)
11	6, 9, 10	ifbieq12d 4510	. . 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 1195	. . . . 5 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → 𝑃 ≈ 2o)
14		relen 8900	. . . . . 6 ⊢ Rel ≈
15	14	brrelex1i 5688	. . . . 5 ⊢ (𝑃 ≈ 2o → 𝑃 ∈ V)
16		difexg 5276	. . . . 5 ⊢ (𝑃 ∈ V → (𝑃 ∖ {𝑍}) ∈ V)
17		uniexg 7695	. . . . 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 4531	. . . 4 ⊢ ((∪ (𝑃 ∖ {𝑍}) ∈ V ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
20	18, 19	sylancom 589	. . 3 ⊢ (((𝐷 ∈ 𝑉 ∧ 𝑃 ⊆ 𝐷 ∧ 𝑃 ≈ 2o) ∧ 𝑍 ∈ 𝐷) → if(𝑍 ∈ 𝑃, ∪ (𝑃 ∖ {𝑍}), 𝑍) ∈ V)
21	5, 11, 12, 20	fvmptd3 6973	. 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 2772	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 1087   = wceq 1542   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ifcif 4481  {csn 4582  ∪ cuni 4865   class class class wbr 5100   ↦ cmpt 5181  ‘cfv 6500  2_oc2o 8401   ≈ cen 8892  pmTrspcpmtr 19382

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-en 8896  df-pmtr 19383

This theorem is referenced by:  pmtrprfv  19394  pmtrprfv3  19395  pmtrmvd  19397  pmtrffv  19400