Theorem pmtridf1o 33070

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 4480	. . . . 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 2780	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5		f1oi 6806	. . . 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 6756	. . . 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 2934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
12		iffalse 4483	. . . . . 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 2780	. . . 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 4773	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22		enpr2 9902	. . . . . 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 2733	. . . . . 6 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25		eqid 2733	. . . . . 6 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
26	24, 25	pmtrrn 19371	. . . . 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 2833	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
29	24, 25	pmtrff1o 19377	. . 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 3016	1 ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929   ⊆ wss 3898  ifcif 4474  {cpr 4577   class class class wbr 5093   I cid 5513  ran crn 5620   ↾ cres 5621  –1-1-onto→wf1o 6485  ‘cfv 6486  2_oc2o 8385   ≈ cen 8872  pmTrspcpmtr 19355

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-iun 4943  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-om 7803  df-1o 8391  df-2o 8392  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-pmtr 19356

This theorem is referenced by:  reprpmtf1o  34660  hgt750lema  34691