Theorem pmtridfv2 33089

Description: Value at Y 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
pmtridfv2	⊢ (𝜑 → (𝑇‘𝑌) = 𝑋)

Proof of Theorem pmtridfv2

Step	Hyp	Ref	Expression
1		pmtridf1o.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
2		fvresi 7207	. . . . 5 ⊢ (𝑌 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
4	3	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌)
5		pmtridf1o.t	. . . . 5 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
6		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
7	6	iftrued 4556	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
8	5, 7	eqtrid 2792	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
9	8	fveq1d 6922	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = (( I ↾ 𝐴)‘𝑌))
10	4, 9, 6	3eqtr4d 2790	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = 𝑋)
11		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
12	11	neneqd 2951	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
13	12	iffalsed 4559	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
14	5, 13	eqtrid 2792	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15	14	fveq1d 6922	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌))
16		pmtridf1o.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉)
18		pmtridf1o.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
19	18	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
20	1	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
21		eqid 2740	. . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
22	21	pmtrprfv2 33081	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
23	17, 19, 20, 11, 22	syl13anc 1372	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
24	15, 23	eqtrd 2780	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = 𝑋)
25	10, 24	pm2.61dane 3035	1 ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946  ifcif 4548  {cpr 4650   I cid 5592   ↾ cres 5702  ‘cfv 6573  pmTrspcpmtr 19483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1o 8522  df-2o 8523  df-en 9004  df-pmtr 19484

This theorem is referenced by:  reprpmtf1o  34603