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Theorem endjusym 6984
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 6946 . . . . . . . . 9 inl:V–1-1-onto→({∅} × V)
2 f1of1 5369 . . . . . . . . 9 (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
31, 2ax-mp 5 . . . . . . . 8 inl:V–1-1→({∅} × V)
4 ssv 3119 . . . . . . . 8 𝐴 ⊆ V
5 f1ores 5385 . . . . . . . 8 ((inl:V–1-1→({∅} × V) ∧ 𝐴 ⊆ V) → (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴))
63, 4, 5mp2an 422 . . . . . . 7 (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴)
7 f1oeng 6654 . . . . . . 7 ((𝐴𝑉 ∧ (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴)) → 𝐴 ≈ (inl “ 𝐴))
86, 7mpan2 421 . . . . . 6 (𝐴𝑉𝐴 ≈ (inl “ 𝐴))
98ensymd 6680 . . . . 5 (𝐴𝑉 → (inl “ 𝐴) ≈ 𝐴)
10 djurf1o 6947 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
11 f1of1 5369 . . . . . . . 8 (inr:V–1-1-onto→({1o} × V) → inr:V–1-1→({1o} × V))
1210, 11ax-mp 5 . . . . . . 7 inr:V–1-1→({1o} × V)
13 f1ores 5385 . . . . . . 7 ((inr:V–1-1→({1o} × V) ∧ 𝐴 ⊆ V) → (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴))
1412, 4, 13mp2an 422 . . . . . 6 (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴)
15 f1oeng 6654 . . . . . 6 ((𝐴𝑉 ∧ (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴)) → 𝐴 ≈ (inr “ 𝐴))
1614, 15mpan2 421 . . . . 5 (𝐴𝑉𝐴 ≈ (inr “ 𝐴))
17 entr 6681 . . . . 5 (((inl “ 𝐴) ≈ 𝐴𝐴 ≈ (inr “ 𝐴)) → (inl “ 𝐴) ≈ (inr “ 𝐴))
189, 16, 17syl2anc 408 . . . 4 (𝐴𝑉 → (inl “ 𝐴) ≈ (inr “ 𝐴))
1918adantr 274 . . 3 ((𝐴𝑉𝐵𝑊) → (inl “ 𝐴) ≈ (inr “ 𝐴))
20 ssv 3119 . . . . . . . 8 𝐵 ⊆ V
21 f1ores 5385 . . . . . . . 8 ((inr:V–1-1→({1o} × V) ∧ 𝐵 ⊆ V) → (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵))
2212, 20, 21mp2an 422 . . . . . . 7 (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵)
23 f1oeng 6654 . . . . . . 7 ((𝐵𝑊 ∧ (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵)) → 𝐵 ≈ (inr “ 𝐵))
2422, 23mpan2 421 . . . . . 6 (𝐵𝑊𝐵 ≈ (inr “ 𝐵))
2524adantl 275 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐵 ≈ (inr “ 𝐵))
2625ensymd 6680 . . . 4 ((𝐴𝑉𝐵𝑊) → (inr “ 𝐵) ≈ 𝐵)
27 f1ores 5385 . . . . . . 7 ((inl:V–1-1→({∅} × V) ∧ 𝐵 ⊆ V) → (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵))
283, 20, 27mp2an 422 . . . . . 6 (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵)
29 f1oeng 6654 . . . . . 6 ((𝐵𝑊 ∧ (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵)) → 𝐵 ≈ (inl “ 𝐵))
3028, 29mpan2 421 . . . . 5 (𝐵𝑊𝐵 ≈ (inl “ 𝐵))
3130adantl 275 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐵 ≈ (inl “ 𝐵))
32 entr 6681 . . . 4 (((inr “ 𝐵) ≈ 𝐵𝐵 ≈ (inl “ 𝐵)) → (inr “ 𝐵) ≈ (inl “ 𝐵))
3326, 31, 32syl2anc 408 . . 3 ((𝐴𝑉𝐵𝑊) → (inr “ 𝐵) ≈ (inl “ 𝐵))
34 djuin 6952 . . . 4 ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
3534a1i 9 . . 3 ((𝐴𝑉𝐵𝑊) → ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅)
36 incom 3268 . . . . 5 ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ((inr “ 𝐴) ∩ (inl “ 𝐵))
37 djuin 6952 . . . . 5 ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ∅
3836, 37eqtr3i 2162 . . . 4 ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅
3938a1i 9 . . 3 ((𝐴𝑉𝐵𝑊) → ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)
40 unen 6713 . . 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 1217 . 2 ((𝐴𝑉𝐵𝑊) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
42 djuun 6955 . 2 ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴𝐵)
43 uncom 3220 . . 3 ((inr “ 𝐴) ∪ (inl “ 𝐵)) = ((inl “ 𝐵) ∪ (inr “ 𝐴))
44 djuun 6955 . . 3 ((inl “ 𝐵) ∪ (inr “ 𝐴)) = (𝐵𝐴)
4543, 44eqtri 2160 . 2 ((inr “ 𝐴) ∪ (inl “ 𝐵)) = (𝐵𝐴)
4641, 42, 453brtr3g 3964 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≈ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  Vcvv 2686  cun 3069  cin 3070  wss 3071  c0 3363  {csn 3527   class class class wbr 3932   × cxp 4540  cres 4544  cima 4545  1-1wf1 5123  1-1-ontowf1o 5125  1oc1o 6309  cen 6635  cdju 6925  inlcinl 6933  inrcinr 6934
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4046  ax-sep 4049  ax-nul 4057  ax-pow 4101  ax-pr 4134  ax-un 4358
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3740  df-iun 3818  df-br 3933  df-opab 3993  df-mpt 3994  df-tr 4030  df-id 4218  df-iord 4291  df-on 4293  df-suc 4296  df-xp 4548  df-rel 4549  df-cnv 4550  df-co 4551  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-iota 5091  df-fun 5128  df-fn 5129  df-f 5130  df-f1 5131  df-fo 5132  df-f1o 5133  df-fv 5134  df-1st 6041  df-2nd 6042  df-1o 6316  df-er 6432  df-en 6638  df-dju 6926  df-inl 6935  df-inr 6936
This theorem is referenced by:  sbthom  13369
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