Theorem pmtridf1o 33110

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 4511	. . . . 5 ⊢ (𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
3	2	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
4	1, 3	eqtrid 2783	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5		f1oi 6861	. . . 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 6811	. . . 4 ⊢ (𝑇 = ( I ↾ 𝐴) → (𝑇:𝐴–1-1-onto→𝐴 ↔ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴))
8	7	biimpar 477	. . 3 ⊢ ((𝑇 = ( I ↾ 𝐴) ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝑇:𝐴–1-1-onto→𝐴)
9	4, 6, 8	syl2anc 584	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇:𝐴–1-1-onto→𝐴)
10		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
11	10	neneqd 2938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
12		iffalse 4514	. . . . . 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	eqtrid 2783	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15		pmtridf1o.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
16	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉)
17		pmtridf1o.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
18	17	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
19		pmtridf1o.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
20	19	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
21	18, 20	prssd 4803	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22		enpr2 10021	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ≈ 2o)
23	18, 20, 10, 22	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ≈ 2o)
24		eqid 2736	. . . . . 6 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25		eqid 2736	. . . . . 6 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
26	24, 25	pmtrrn 19443	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑋, 𝑌} ⊆ 𝐴 ∧ {𝑋, 𝑌} ≈ 2o) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
27	16, 21, 23, 26	syl3anc 1373	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
28	14, 27	eqeltrd 2835	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
29	24, 25	pmtrff1o 19449	. . 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 3020	1 ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2933   ⊆ wss 3931  ifcif 4505  {cpr 4608   class class class wbr 5124   I cid 5552  ran crn 5660   ↾ cres 5661  –1-1-onto→wf1o 6535  ‘cfv 6536  2_oc2o 8479   ≈ cen 8961  pmTrspcpmtr 19427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-om 7867  df-1o 8485  df-2o 8486  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-pmtr 19428

This theorem is referenced by:  reprpmtf1o  34663  hgt750lema  34694