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Theorem endjusym 7109
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 7071 . . . . . . . . 9 inl:V–1-1-onto→({∅} × V)
2 f1of1 5472 . . . . . . . . 9 (inl:V–1-1-onto→({∅} × V) → inl:V–1-1→({∅} × V))
31, 2ax-mp 5 . . . . . . . 8 inl:V–1-1→({∅} × V)
4 ssv 3189 . . . . . . . 8 𝐴 ⊆ V
5 f1ores 5488 . . . . . . . 8 ((inl:V–1-1→({∅} × V) ∧ 𝐴 ⊆ V) → (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴))
63, 4, 5mp2an 426 . . . . . . 7 (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴)
7 f1oeng 6771 . . . . . . 7 ((𝐴𝑉 ∧ (inl ↾ 𝐴):𝐴1-1-onto→(inl “ 𝐴)) → 𝐴 ≈ (inl “ 𝐴))
86, 7mpan2 425 . . . . . 6 (𝐴𝑉𝐴 ≈ (inl “ 𝐴))
98ensymd 6797 . . . . 5 (𝐴𝑉 → (inl “ 𝐴) ≈ 𝐴)
10 djurf1o 7072 . . . . . . . 8 inr:V–1-1-onto→({1o} × V)
11 f1of1 5472 . . . . . . . 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 5488 . . . . . . 7 ((inr:V–1-1→({1o} × V) ∧ 𝐴 ⊆ V) → (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴))
1412, 4, 13mp2an 426 . . . . . 6 (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴)
15 f1oeng 6771 . . . . . 6 ((𝐴𝑉 ∧ (inr ↾ 𝐴):𝐴1-1-onto→(inr “ 𝐴)) → 𝐴 ≈ (inr “ 𝐴))
1614, 15mpan2 425 . . . . 5 (𝐴𝑉𝐴 ≈ (inr “ 𝐴))
17 entr 6798 . . . . 5 (((inl “ 𝐴) ≈ 𝐴𝐴 ≈ (inr “ 𝐴)) → (inl “ 𝐴) ≈ (inr “ 𝐴))
189, 16, 17syl2anc 411 . . . 4 (𝐴𝑉 → (inl “ 𝐴) ≈ (inr “ 𝐴))
1918adantr 276 . . 3 ((𝐴𝑉𝐵𝑊) → (inl “ 𝐴) ≈ (inr “ 𝐴))
20 ssv 3189 . . . . . . . 8 𝐵 ⊆ V
21 f1ores 5488 . . . . . . . 8 ((inr:V–1-1→({1o} × V) ∧ 𝐵 ⊆ V) → (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵))
2212, 20, 21mp2an 426 . . . . . . 7 (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵)
23 f1oeng 6771 . . . . . . 7 ((𝐵𝑊 ∧ (inr ↾ 𝐵):𝐵1-1-onto→(inr “ 𝐵)) → 𝐵 ≈ (inr “ 𝐵))
2422, 23mpan2 425 . . . . . 6 (𝐵𝑊𝐵 ≈ (inr “ 𝐵))
2524adantl 277 . . . . 5 ((𝐴𝑉𝐵𝑊) → 𝐵 ≈ (inr “ 𝐵))
2625ensymd 6797 . . . 4 ((𝐴𝑉𝐵𝑊) → (inr “ 𝐵) ≈ 𝐵)
27 f1ores 5488 . . . . . . 7 ((inl:V–1-1→({∅} × V) ∧ 𝐵 ⊆ V) → (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵))
283, 20, 27mp2an 426 . . . . . 6 (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵)
29 f1oeng 6771 . . . . . 6 ((𝐵𝑊 ∧ (inl ↾ 𝐵):𝐵1-1-onto→(inl “ 𝐵)) → 𝐵 ≈ (inl “ 𝐵))
3028, 29mpan2 425 . . . . 5 (𝐵𝑊𝐵 ≈ (inl “ 𝐵))
3130adantl 277 . . . 4 ((𝐴𝑉𝐵𝑊) → 𝐵 ≈ (inl “ 𝐵))
32 entr 6798 . . . 4 (((inr “ 𝐵) ≈ 𝐵𝐵 ≈ (inl “ 𝐵)) → (inr “ 𝐵) ≈ (inl “ 𝐵))
3326, 31, 32syl2anc 411 . . 3 ((𝐴𝑉𝐵𝑊) → (inr “ 𝐵) ≈ (inl “ 𝐵))
34 djuin 7077 . . . 4 ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅
3534a1i 9 . . 3 ((𝐴𝑉𝐵𝑊) → ((inl “ 𝐴) ∩ (inr “ 𝐵)) = ∅)
36 incom 3339 . . . . 5 ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ((inr “ 𝐴) ∩ (inl “ 𝐵))
37 djuin 7077 . . . . 5 ((inl “ 𝐵) ∩ (inr “ 𝐴)) = ∅
3836, 37eqtr3i 2210 . . . 4 ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅
3938a1i 9 . . 3 ((𝐴𝑉𝐵𝑊) → ((inr “ 𝐴) ∩ (inl “ 𝐵)) = ∅)
40 unen 6830 . . 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 1249 . 2 ((𝐴𝑉𝐵𝑊) → ((inl “ 𝐴) ∪ (inr “ 𝐵)) ≈ ((inr “ 𝐴) ∪ (inl “ 𝐵)))
42 djuun 7080 . 2 ((inl “ 𝐴) ∪ (inr “ 𝐵)) = (𝐴𝐵)
43 uncom 3291 . . 3 ((inr “ 𝐴) ∪ (inl “ 𝐵)) = ((inl “ 𝐵) ∪ (inr “ 𝐴))
44 djuun 7080 . . 3 ((inl “ 𝐵) ∪ (inr “ 𝐴)) = (𝐵𝐴)
4543, 44eqtri 2208 . 2 ((inr “ 𝐴) ∪ (inl “ 𝐵)) = (𝐵𝐴)
4641, 42, 453brtr3g 4048 1 ((𝐴𝑉𝐵𝑊) → (𝐴𝐵) ≈ (𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1363  wcel 2158  Vcvv 2749  cun 3139  cin 3140  wss 3141  c0 3434  {csn 3604   class class class wbr 4015   × cxp 4636  cres 4640  cima 4641  1-1wf1 5225  1-1-ontowf1o 5227  1oc1o 6424  cen 6752  cdju 7050  inlcinl 7058  inrcinr 7059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-1st 6155  df-2nd 6156  df-1o 6431  df-er 6549  df-en 6755  df-dju 7051  df-inl 7060  df-inr 7061
This theorem is referenced by:  sbthom  15071
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