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Theorem endjusym 6931
Description: Reversing right and left operands of a disjoint union produces an equinumerous result. (Contributed by Jim Kingdon, 10-Jul-2023.)
Assertion
Ref Expression
endjusym ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≈ (𝐵𝐴))

Proof of Theorem endjusym
StepHypRef Expression
1 djulf1o 6893 . . . . . . . . 9 inl:V–1-1-onto→({∅} × V)
2 f1of1 5320 . . . . . . . . 9 (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
31, 2ax-mp 7 . . . . . . . 8 inl:V–1-1→({∅} × V)
4 ssv 3083 . . . . . . . 8 𝐴 ⊆ V
5 f1ores 5336 . . . . . . . 8 ((inl:V–1-1→({∅} × V) ∧ 𝐴 ⊆ V) → (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴))
63, 4, 5mp2an 420 . . . . . . 7 (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴)
7 f1oeng 6603 . . . . . . 7 ((𝐴𝑉 ∧ (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴)) → 𝐴 ≈ (inl “ 𝐴))
86, 7mpan2 419 . . . . . 6 (𝐴𝑉𝐴 ≈ (inl “ 𝐴))
98ensymd 6629 . . . . 5 (𝐴𝑉 → (inl “ 𝐴) ≈ 𝐴)
10 djurf1o 6894 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
11 f1of1 5320 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → inr:V–1-1→({1o} × V))
1210, 11ax-mp 7 . . . . . . 7 inr:V–1-1→({1o} × V)
13 f1ores 5336 . . . . . . 7 ((inr:V–1-1→({1o} × V) ∧ 𝐴 ⊆ V) → (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴))
1412, 4, 13mp2an 420 . . . . . 6 (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴)
15 f1oeng 6603 . . . . . 6 ((𝐴𝑉 ∧ (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴)) → 𝐴 ≈ (inr “ 𝐴))
1614, 15mpan2 419 . . . . 5 (𝐴𝑉𝐴 ≈ (inr “ 𝐴))
17 entr 6630 . . . . 5 (((inl “ 𝐴) ≈ 𝐴𝐴 ≈ (inr “ 𝐴)) → (inl “ 𝐴) ≈ (inr “ 𝐴))
189, 16, 17syl2anc 406 . . . 4 (𝐴𝑉 → (inl “ 𝐴) ≈ (inr “ 𝐴))
1918adantr 272 . . 3 ((𝐴𝑉𝐵𝑊) → (inl “ 𝐴) ≈ (inr “ 𝐴))
20 ssv 3083 . . . . . . . 8 𝐵 ⊆ V
21 f1ores 5336 . . . . . . . 8 ((inr:V–1-1→({1o} × V) ∧ 𝐵 ⊆ V) → (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵))
2212, 20, 21mp2an 420 . . . . . . 7 (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵)
23 f1oeng 6603 . . . . . . 7 ((𝐵𝑊 ∧ (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵)) → 𝐵 ≈ (inr “ 𝐵))
2422, 23mpan2 419 . . . . . 6 (𝐵𝑊𝐵 ≈ (inr “ 𝐵))
2524adantl 273 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐵 ≈ (inr “ 𝐵))
2625ensymd 6629 . . . 4 ((𝐴𝑉𝐵𝑊) → (inr “ 𝐵) ≈ 𝐵)
27 f1ores 5336 . . . . . . 7 ((inl:V–1-1→({∅} × V) ∧ 𝐵 ⊆ V) → (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵))
283, 20, 27mp2an 420 . . . . . 6 (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵)
29 f1oeng 6603 . . . . . 6 ((𝐵𝑊 ∧ (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵)) → 𝐵 ≈ (inl “ 𝐵))
3028, 29mpan2 419 . . . . 5 (𝐵𝑊𝐵 ≈ (inl “ 𝐵))
3130adantl 273 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐵 ≈ (inl “ 𝐵))
32 entr 6630 . . . 4 (((inr “ 𝐵) ≈ 𝐵𝐵 ≈ (inl “ 𝐵)) → (inr “ 𝐵) ≈ (inl “ 𝐵))
3326, 31, 32syl2anc 406 . . 3 ((𝐴𝑉𝐵𝑊) → (inr “ 𝐵) ≈ (inl “ 𝐵))
34 djuin 6899 . . . 4 ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
3534a1i 9 . . 3 ((𝐴𝑉𝐵𝑊) → ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅)
36 incom 3232 . . . . 5 ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ((inr “ 𝐴) ∩ (inl “ 𝐵))
37 djuin 6899 . . . . 5 ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ∅
3836, 37eqtr3i 2135 . . . 4 ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅
3938a1i 9 . . 3 ((𝐴𝑉𝐵𝑊) → ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)
40 unen 6662 . . 3 ((((inl “ 𝐴) ≈ (inr “ 𝐴) ∧ (inr “ 𝐵) ≈ (inl “ 𝐵)) ∧ (((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅ ∧ ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
4119, 33, 35, 39, 40syl22anc 1198 . 2 ((𝐴𝑉𝐵𝑊) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
42 djuun 6902 . 2 ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴𝐵)
43 uncom 3184 . . 3 ((inr “ 𝐴) ∪ (inl “ 𝐵)) = ((inl “ 𝐵) ∪ (inr “ 𝐴))
44 djuun 6902 . . 3 ((inl “ 𝐵) ∪ (inr “ 𝐴)) = (𝐵𝐴)
4543, 44eqtri 2133 . 2 ((inr “ 𝐴) ∪ (inl “ 𝐵)) = (𝐵𝐴)
4641, 42, 453brtr3g 3924 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≈ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1312  wcel 1461  Vcvv 2655  cun 3033  cin 3034  wss 3035  c0 3327  {csn 3491   class class class wbr 3893   × cxp 4495  cres 4499  cima 4500  1-1wf1 5076  1-1-ontowf1o 5078  1oc1o 6258  cen 6584  cdju 6872  inlcinl 6880  inrcinr 6881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-coll 4001  ax-sep 4004  ax-nul 4012  ax-pow 4056  ax-pr 4089  ax-un 4313
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-csb 2970  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-nul 3328  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-iun 3779  df-br 3894  df-opab 3948  df-mpt 3949  df-tr 3985  df-id 4173  df-iord 4246  df-on 4248  df-suc 4251  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-rn 4508  df-res 4509  df-ima 4510  df-iota 5044  df-fun 5081  df-fn 5082  df-f 5083  df-f1 5084  df-fo 5085  df-f1o 5086  df-fv 5087  df-1st 5990  df-2nd 5991  df-1o 6265  df-er 6381  df-en 6587  df-dju 6873  df-inl 6882  df-inr 6883
This theorem is referenced by:  sbthom  12902
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