![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > pmtridf1o | Structured version Visualization version GIF version |
Description: Transpositions of 𝑋 and 𝑌 (understood to be the identity when 𝑋 = 𝑌), are bijections. (Contributed by Thierry Arnoux, 1-Jan-2022.) |
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})) |
Ref | Expression |
---|---|
pmtridf1o | ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pmtridf1o.t | . . . 4 ⊢ 𝑇 = if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) | |
2 | iftrue 4493 | . . . . 5 ⊢ (𝑋 = 𝑌 → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴)) | |
3 | 2 | adantl 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → if(𝑋 = 𝑌, ( I ↾ 𝐴), ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) = ( I ↾ 𝐴)) |
4 | 1, 3 | eqtrid 2785 | . . 3 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇 = ( I ↾ 𝐴)) |
5 | f1oi 6823 | . . . 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 6773 | . . . 4 ⊢ (𝑇 = ( I ↾ 𝐴) → (𝑇:𝐴–1-1-onto→𝐴 ↔ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴)) | |
8 | 7 | biimpar 479 | . . 3 ⊢ ((𝑇 = ( I ↾ 𝐴) ∧ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴) → 𝑇:𝐴–1-1-onto→𝐴) |
9 | 4, 6, 8 | syl2anc 585 | . 2 ⊢ ((𝜑 ∧ 𝑋 = 𝑌) → 𝑇:𝐴–1-1-onto→𝐴) |
10 | simpr 486 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ≠ 𝑌) | |
11 | 10 | neneqd 2945 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑋 = 𝑌) |
12 | iffalse 4496 | . . . . . 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 2785 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 = ((pmTrsp‘𝐴)‘{𝑋, 𝑌})) |
15 | pmtridf1o.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
16 | 15 | adantr 482 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝐴 ∈ 𝑉) |
17 | pmtridf1o.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
18 | 17 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑋 ∈ 𝐴) |
19 | pmtridf1o.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
20 | 19 | adantr 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑌 ∈ 𝐴) |
21 | 18, 20 | prssd 4783 | . . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ⊆ 𝐴) |
22 | enpr2 9943 | . . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑋 ≠ 𝑌) → {𝑋, 𝑌} ≈ 2o) | |
23 | 18, 20, 10, 22 | syl3anc 1372 | . . . . 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 19244 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ {𝑋, 𝑌} ⊆ 𝐴 ∧ {𝑋, 𝑌} ≈ 2o) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴)) |
27 | 16, 21, 23, 26 | syl3anc 1372 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → ((pmTrsp‘𝐴)‘{𝑋, 𝑌}) ∈ ran (pmTrsp‘𝐴)) |
28 | 14, 27 | eqeltrd 2834 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ 𝑌) → 𝑇 ∈ ran (pmTrsp‘𝐴)) |
29 | 24, 25 | pmtrff1o 19250 | . . 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 3029 | 1 ⊢ (𝜑 → 𝑇:𝐴–1-1-onto→𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ⊆ wss 3911 ifcif 4487 {cpr 4589 class class class wbr 5106 I cid 5531 ran crn 5635 ↾ cres 5636 –1-1-onto→wf1o 6496 ‘cfv 6497 2oc2o 8407 ≈ cen 8883 pmTrspcpmtr 19228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-om 7804 df-1o 8413 df-2o 8414 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-pmtr 19229 |
This theorem is referenced by: reprpmtf1o 33296 hgt750lema 33327 |
Copyright terms: Public domain | W3C validator |