Theorem pmtridfv2 33703

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 7178	. . . . 5 ⊢ (𝑌 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
3	1, 2	syl 18	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑌) = 𝑌)
4	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑌) = 𝑌)
5		pmtridf1o.t	. . . . 5 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
6		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
7	6	iftrued 4489	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
8	5, 7	eqtrid 2807	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
9	8	fveq1d 6887	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = (( I ↾ 𝐴)‘𝑌))
10	4, 9, 6	3eqtr4d 2805	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑌) = 𝑋)
11		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
12	11	neneqd 2960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
13	12	iffalsed 4492	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
14	5, 13	eqtrid 2807	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15	14	fveq1d 6887	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌))
16		pmtridf1o.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	16	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉)
18		pmtridf1o.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
19	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
20	1	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
21		eqid 2760	. . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
22	21	pmtrprfv2 33695	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
23	17, 19, 20, 11, 22	syl13anc 1399	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑌) = 𝑋)
24	15, 23	eqtrd 2795	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑌) = 𝑋)
25	10, 24	pm2.61dane 3042	1 ⊢ (𝜑 → (𝑇‘𝑌) = 𝑋)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ifcif 4481  {cpr 4586   I cid 5544   ↾ cres 5652  ‘cfv 6538  pmTrspcpmtr 19660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-suc 6368  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-1o 8479  df-2o 8480  df-en 8977  df-pmtr 19661

This theorem is used by:  reprpmtf1o  35301