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Theorem undifdc 6812
Description: Union of complementary parts into whole. This is a case where we can strengthen undifss 3443 from subset to equality. (Contributed by Jim Kingdon, 17-Jun-2022.)
Assertion
Ref Expression
undifdc ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem undifdc
Dummy variables 𝑣 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4 (𝑤 = ∅ → 𝑤 = ∅)
2 difeq2 3188 . . . 4 (𝑤 = ∅ → (𝐴𝑤) = (𝐴 ∖ ∅))
31, 2uneq12d 3231 . . 3 (𝑤 = ∅ → (𝑤 ∪ (𝐴𝑤)) = (∅ ∪ (𝐴 ∖ ∅)))
43eqeq2d 2151 . 2 (𝑤 = ∅ → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))))
5 id 19 . . . 4 (𝑤 = 𝑣𝑤 = 𝑣)
6 difeq2 3188 . . . 4 (𝑤 = 𝑣 → (𝐴𝑤) = (𝐴𝑣))
75, 6uneq12d 3231 . . 3 (𝑤 = 𝑣 → (𝑤 ∪ (𝐴𝑤)) = (𝑣 ∪ (𝐴𝑣)))
87eqeq2d 2151 . 2 (𝑤 = 𝑣 → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (𝑣 ∪ (𝐴𝑣))))
9 id 19 . . . 4 (𝑤 = (𝑣 ∪ {𝑧}) → 𝑤 = (𝑣 ∪ {𝑧}))
10 difeq2 3188 . . . 4 (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴𝑤) = (𝐴 ∖ (𝑣 ∪ {𝑧})))
119, 10uneq12d 3231 . . 3 (𝑤 = (𝑣 ∪ {𝑧}) → (𝑤 ∪ (𝐴𝑤)) = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
1211eqeq2d 2151 . 2 (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
13 id 19 . . . 4 (𝑤 = 𝐵𝑤 = 𝐵)
14 difeq2 3188 . . . 4 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
1513, 14uneq12d 3231 . . 3 (𝑤 = 𝐵 → (𝑤 ∪ (𝐴𝑤)) = (𝐵 ∪ (𝐴𝐵)))
1615eqeq2d 2151 . 2 (𝑤 = 𝐵 → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (𝐵 ∪ (𝐴𝐵))))
17 un0 3396 . . . 4 ((𝐴 ∖ ∅) ∪ ∅) = (𝐴 ∖ ∅)
18 uncom 3220 . . . 4 ((𝐴 ∖ ∅) ∪ ∅) = (∅ ∪ (𝐴 ∖ ∅))
19 dif0 3433 . . . 4 (𝐴 ∖ ∅) = 𝐴
2017, 18, 193eqtr3ri 2169 . . 3 𝐴 = (∅ ∪ (𝐴 ∖ ∅))
2120a1i 9 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (∅ ∪ (𝐴 ∖ ∅)))
22 difundi 3328 . . . . . . 7 (𝐴 ∖ (𝑣 ∪ {𝑧})) = ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧}))
2322uneq2i 3227 . . . . . 6 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = ((𝑣 ∪ {𝑧}) ∪ ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧})))
24 undi 3324 . . . . . 6 ((𝑣 ∪ {𝑧}) ∪ ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
2523, 24eqtri 2160 . . . . 5 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
26 simp3 983 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐵𝐴)
2726ad3antrrr 483 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐵𝐴)
28 simplrr 525 . . . . . . . . . . . 12 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧 ∈ (𝐵𝑣))
2928eldifad 3082 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧𝐵)
3027, 29sseldd 3098 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧𝐴)
3130snssd 3665 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → {𝑧} ⊆ 𝐴)
32 ssequn1 3246 . . . . . . . . 9 ({𝑧} ⊆ 𝐴 ↔ ({𝑧} ∪ 𝐴) = 𝐴)
3331, 32sylib 121 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ 𝐴) = 𝐴)
34 simpr 109 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = (𝑣 ∪ (𝐴𝑣)))
3534uneq2d 3230 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ 𝐴) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣))))
3633, 35eqtr3d 2174 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣))))
37 uncom 3220 . . . . . . . . 9 (𝑣 ∪ {𝑧}) = ({𝑧} ∪ 𝑣)
3837uneq1i 3226 . . . . . . . 8 ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = (({𝑧} ∪ 𝑣) ∪ (𝐴𝑣))
39 unass 3233 . . . . . . . 8 (({𝑧} ∪ 𝑣) ∪ (𝐴𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣)))
4038, 39eqtri 2160 . . . . . . 7 ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣)))
4136, 40syl6reqr 2191 . . . . . 6 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = 𝐴)
42 unass 3233 . . . . . . . 8 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧})))
43 uncom 3220 . . . . . . . . . 10 ({𝑧} ∪ (𝐴 ∖ {𝑧})) = ((𝐴 ∖ {𝑧}) ∪ {𝑧})
44 simp1 981 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
4544ad3antrrr 483 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
46 dcdifsnid 6400 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑧𝐴) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
4745, 30, 46syl2anc 408 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
4843, 47syl5eq 2184 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ (𝐴 ∖ {𝑧})) = 𝐴)
4948uneq2d 3230 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧}))) = (𝑣𝐴))
5042, 49syl5eq 2184 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣𝐴))
51 simplrl 524 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑣𝐵)
5251, 27sstrd 3107 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑣𝐴)
53 ssequn1 3246 . . . . . . . 8 (𝑣𝐴 ↔ (𝑣𝐴) = 𝐴)
5452, 53sylib 121 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (𝑣𝐴) = 𝐴)
5550, 54eqtrd 2172 . . . . . 6 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = 𝐴)
5641, 55ineq12d 3278 . . . . 5 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧}))) = (𝐴𝐴))
5725, 56syl5eq 2184 . . . 4 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (𝐴𝐴))
58 inidm 3285 . . . 4 (𝐴𝐴) = 𝐴
5957, 58syl6req 2189 . . 3 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
6059ex 114 . 2 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) → (𝐴 = (𝑣 ∪ (𝐴𝑣)) → 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
61 simp2 982 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
624, 8, 12, 16, 21, 60, 61findcard2sd 6786 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 819  w3a 962   = wceq 1331  wcel 1480  wral 2416  cdif 3068  cun 3069  cin 3070  wss 3071  c0 3363  {csn 3527  Fincfn 6634
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 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  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-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-er 6429  df-en 6635  df-fin 6637
This theorem is referenced by:  undiffi  6813
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