Theorem pmtridfv2 32860

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 7177	. . . . 5 ⊢ (𝑌 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
4	3	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌)
5		pmtridf1o.t	. . . . 5 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
6		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
7	6	iftrued 4532	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
8	5, 7	eqtrid 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
9	8	fveq1d 6893	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = (( I ↾ 𝐴)‘𝑌))
10	4, 9, 6	3eqtr4d 2775	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = 𝑋)
11		simpr 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
12	11	neneqd 2935	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
13	12	iffalsed 4535	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
14	5, 13	eqtrid 2777	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15	14	fveq1d 6893	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌))
16		pmtridf1o.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	16	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉)
18		pmtridf1o.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
19	18	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
20	1	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
21		eqid 2725	. . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
22	21	pmtrprfv2 32854	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
23	17, 19, 20, 11, 22	syl13anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
24	15, 23	eqtrd 2765	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = 𝑋)
25	10, 24	pm2.61dane 3019	1 ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  ifcif 4524  {cpr 4626   I cid 5569   ↾ cres 5674  ‘cfv 6542  pmTrspcpmtr 19398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-suc 6370  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1o 8483  df-2o 8484  df-en 8961  df-pmtr 19399

This theorem is referenced by:  reprpmtf1o  34314