Theorem pmtridfv1 30787

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 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑋 = 𝑌)
3	2	iftrued 4433	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
4	1, 3	syl5eq 2845	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5	4	fveq1d 6647	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑋) = (( I ↾ 𝐴)‘𝑋))
6		pmtridf1o.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
7		fvresi 6912	. . . . 5 ⊢ (𝑋 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
8	6, 7	syl 17	. . . 4 ⊢ (𝜑 → (( I ↾ 𝐴)‘𝑋) = 𝑋)
9	8	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (( I ↾ 𝐴)‘𝑋) = 𝑋)
10	5, 9, 2	3eqtrd 2837	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → (𝑇‘𝑋) = 𝑌)
11		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
12	11	neneqd 2992	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
13	12	iffalsed 4436	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
14	1, 13	syl5eq 2845	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15	14	fveq1d 6647	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑋) = (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋))
16		pmtridf1o.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
17	16	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉)
18	6	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
19		pmtridf1o.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
20	19	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
21		eqid 2798	. . . . 5 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
22	21	pmtrprfv 18573	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌)) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
23	17, 18, 20, 11, 22	syl13anc 1369	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (((pmTrsp‘𝐴)‘{𝑋, 𝑌})‘𝑋) = 𝑌)
24	15, 23	eqtrd 2833	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → (𝑇‘𝑋) = 𝑌)
25	10, 24	pm2.61dane 3074	1 ⊢ (𝜑 → (𝑇‘𝑋) = 𝑌)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ifcif 4425  {cpr 4527   I cid 5424   ↾ cres 5521  ‘cfv 6324  pmTrspcpmtr 18561

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-1o 8085  df-2o 8086  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pmtr 18562

This theorem is referenced by: (None)