Theorem pmtridfv1 33071

Description: Value at X of the transposition of 𝑋 and 𝑌 (understood to be the identity when X = Y ). (Contributed by Thierry Arnoux, 3-Jan-2022.)

Hypotheses

Ref	Expression
pmtridf1o.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
pmtridf1o.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)
pmtridf1o.y	⊢ (𝜑 → 𝑌 ∈ 𝐴)
pmtridf1o.t	⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))

Assertion

Ref	Expression
pmtridfv1	⊢ (𝜑 → (𝑇‘𝑋) = 𝑌)

Proof of Theorem pmtridfv1

Step	Hyp	Ref	Expression
1		pmtridf1o.t	. . . . 5 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
2		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
3	2	iftrued 4482	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
4	1, 3	eqtrid 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5	4	fveq1d 6830	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑋) = (( I ↾ 𝐴)‘𝑋))
6		pmtridf1o.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
7		fvresi 7113	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
9	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑋) = 𝑋)
10	5, 9, 2	3eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑋) = 𝑌)
11		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
12	11	neneqd 2934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
13	12	iffalsed 4485	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
14	1, 13	eqtrid 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15	14	fveq1d 6830	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑋) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋))
16		pmtridf1o.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉)
18	6	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
19		pmtridf1o.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
21		eqid 2733	. . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
22	21	pmtrprfv 19367	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
23	17, 18, 20, 11, 22	syl13anc 1374	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
24	15, 23	eqtrd 2768	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑋) = 𝑌)
25	10, 24	pm2.61dane 3016	1 ⊢ (𝜑 → (𝑇‘𝑋) = 𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ifcif 4474  {cpr 4577   I cid 5513   ↾ cres 5621  ‘cfv 6486  pmTrspcpmtr 19355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-1o 8391  df-2o 8392  df-en 8876  df-pmtr 19356

This theorem is referenced by: (None)