Theorem pmtridf1o 30738

Description: Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-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
pmtridf1o	⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴)

Proof of Theorem pmtridf1o

Step	Hyp	Ref	Expression
1		pmtridf1o.t	. . . 4 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
2		iftrue 4475	. . . . 5 ⊢ (𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
3	2	adantl 484	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
4	1, 3	syl5eq 2870	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5		f1oi 6654	. . . 4 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → ( I ↾ 𝐴):𝐴–1-1-onto→𝐴)
7		f1oeq1 6606	. . . 4 ⊢ (𝑇 = ( I ↾ 𝐴) → (𝑇:𝐴–1-1-onto→𝐴 ↔ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴))
8	7	biimpar 480	. . 3 ⊢ ((𝑇 = ( I ↾ 𝐴) ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝑇:𝐴–1-1-onto→𝐴)
9	4, 6, 8	syl2anc 586	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇:𝐴–1-1-onto→𝐴)
10		simpr 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
11	10	neneqd 3023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
12		iffalse 4478	. . . . . 6 ⊢ (¬ 𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
13	11, 12	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
14	1, 13	syl5eq 2870	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15		pmtridf1o.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
16	15	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉)
17		pmtridf1o.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
18	17	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
19		pmtridf1o.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
20	19	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
21	18, 20	prssd 4757	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22		pr2nelem 9432	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ≈ 2o)
23	18, 20, 10, 22	syl3anc 1367	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ≈ 2o)
24		eqid 2823	. . . . . 6 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25		eqid 2823	. . . . . 6 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
26	24, 25	pmtrrn 18587	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑋, 𝑌} ⊆ 𝐴 ∧ {𝑋, 𝑌} ≈ 2o) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
27	16, 21, 23, 26	syl3anc 1367	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
28	14, 27	eqeltrd 2915	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
29	24, 25	pmtrff1o 18593	. . 3 ⊢ (𝑇 ∈ ran (pmTrsp‘𝐴) → 𝑇:𝐴–1-1-onto→𝐴)
30	28, 29	syl 17	. 2 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇:𝐴–1-1-onto→𝐴)
31	9, 30	pm2.61dane 3106	1 ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   ⊆ wss 3938  ifcif 4469  {cpr 4571   class class class wbr 5068   I cid 5461  ran crn 5558   ↾ cres 5559  –1-1-onto→wf1o 6356  ‘cfv 6357  2_oc2o 8098   ≈ cen 8508  pmTrspcpmtr 18571

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-1o 8104  df-2o 8105  df-er 8291  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-pmtr 18572

This theorem is referenced by:  reprpmtf1o  31899  hgt750lema  31930