Theorem pmtridf1o 30786

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 4431	. . . . 5 ⊢ (𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
3	2	adantl 485	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴))
4	1, 3	syl5eq 2845	. . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴))
5		f1oi 6627	. . . 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 6579	. . . 4 ⊢ (𝑇 = ( I ↾ 𝐴) → (𝑇:𝐴–1-1-onto→𝐴 ↔ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴))
8	7	biimpar 481	. . 3 ⊢ ((𝑇 = ( I ↾ 𝐴) ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝑇:𝐴–1-1-onto→𝐴)
9	4, 6, 8	syl2anc 587	. 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇:𝐴–1-1-onto→𝐴)
10		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌)
11	10	neneqd 2992	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌)
12		iffalse 4434	. . . . . 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 2845	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌}))
15		pmtridf1o.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
16	15	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉)
17		pmtridf1o.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
18	17	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴)
19		pmtridf1o.y	. . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐴)
20	19	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴)
21	18, 20	prssd 4715	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ 𝐴)
22		pr2nelem 9415	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ≈ 2o)
23	18, 20, 10, 22	syl3anc 1368	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ≈ 2o)
24		eqid 2798	. . . . . 6 ⊢ (pmTrsp‘𝐴) = (pmTrsp‘𝐴)
25		eqid 2798	. . . . . 6 ⊢ ran (pmTrsp‘𝐴) = ran (pmTrsp‘𝐴)
26	24, 25	pmtrrn 18577	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑋, 𝑌} ⊆ 𝐴 ∧ {𝑋, 𝑌} ≈ 2o) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
27	16, 21, 23, 26	syl3anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴))
28	14, 27	eqeltrd 2890	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴))
29	24, 25	pmtrff1o 18583	. . 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 3074	1 ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ≠ wne 2987   ⊆ wss 3881  ifcif 4425  {cpr 4527   class class class wbr 5030   I cid 5424  ran crn 5520   ↾ cres 5521  –1-1-onto→wf1o 6323  ‘cfv 6324  2_oc2o 8079   ≈ cen 8489  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-fin 8496  df-pmtr 18562

This theorem is referenced by:  reprpmtf1o  32007  hgt750lema  32038