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Theorem undifdc 6560
Description: Union of complementary parts into whole. This is a case where we can strengthen undifss 3344 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 3096 . . . 4 (𝑤 = ∅ → (𝐴𝑤) = (𝐴 ∖ ∅))
31, 2uneq12d 3139 . . 3 (𝑤 = ∅ → (𝑤 ∪ (𝐴𝑤)) = (∅ ∪ (𝐴 ∖ ∅)))
43eqeq2d 2094 . 2 (𝑤 = ∅ → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))))
5 id 19 . . . 4 (𝑤 = 𝑣𝑤 = 𝑣)
6 difeq2 3096 . . . 4 (𝑤 = 𝑣 → (𝐴𝑤) = (𝐴𝑣))
75, 6uneq12d 3139 . . 3 (𝑤 = 𝑣 → (𝑤 ∪ (𝐴𝑤)) = (𝑣 ∪ (𝐴𝑣)))
87eqeq2d 2094 . 2 (𝑤 = 𝑣 → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (𝑣 ∪ (𝐴𝑣))))
9 id 19 . . . 4 (𝑤 = (𝑣 ∪ {𝑧}) → 𝑤 = (𝑣 ∪ {𝑧}))
10 difeq2 3096 . . . 4 (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴𝑤) = (𝐴 ∖ (𝑣 ∪ {𝑧})))
119, 10uneq12d 3139 . . 3 (𝑤 = (𝑣 ∪ {𝑧}) → (𝑤 ∪ (𝐴𝑤)) = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
1211eqeq2d 2094 . 2 (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
13 id 19 . . . 4 (𝑤 = 𝐵𝑤 = 𝐵)
14 difeq2 3096 . . . 4 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
1513, 14uneq12d 3139 . . 3 (𝑤 = 𝐵 → (𝑤 ∪ (𝐴𝑤)) = (𝐵 ∪ (𝐴𝐵)))
1615eqeq2d 2094 . 2 (𝑤 = 𝐵 → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (𝐵 ∪ (𝐴𝐵))))
17 un0 3299 . . . 4 ((𝐴 ∖ ∅) ∪ ∅) = (𝐴 ∖ ∅)
18 uncom 3128 . . . 4 ((𝐴 ∖ ∅) ∪ ∅) = (∅ ∪ (𝐴 ∖ ∅))
19 dif0 3335 . . . 4 (𝐴 ∖ ∅) = 𝐴
2017, 18, 193eqtr3ri 2112 . . 3 𝐴 = (∅ ∪ (𝐴 ∖ ∅))
2120a1i 9 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (∅ ∪ (𝐴 ∖ ∅)))
22 difundi 3234 . . . . . . 7 (𝐴 ∖ (𝑣 ∪ {𝑧})) = ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧}))
2322uneq2i 3135 . . . . . 6 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = ((𝑣 ∪ {𝑧}) ∪ ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧})))
24 undi 3230 . . . . . 6 ((𝑣 ∪ {𝑧}) ∪ ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
2523, 24eqtri 2103 . . . . 5 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
26 simp3 941 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐵𝐴)
2726ad3antrrr 476 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐵𝐴)
28 simplrr 503 . . . . . . . . . . . 12 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧 ∈ (𝐵𝑣))
2928eldifad 2995 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧𝐵)
3027, 29sseldd 3011 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧𝐴)
3130snssd 3556 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → {𝑧} ⊆ 𝐴)
32 ssequn1 3154 . . . . . . . . 9 ({𝑧} ⊆ 𝐴 ↔ ({𝑧} ∪ 𝐴) = 𝐴)
3331, 32sylib 120 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ 𝐴) = 𝐴)
34 simpr 108 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = (𝑣 ∪ (𝐴𝑣)))
3534uneq2d 3138 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ 𝐴) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣))))
3633, 35eqtr3d 2117 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣))))
37 uncom 3128 . . . . . . . . 9 (𝑣 ∪ {𝑧}) = ({𝑧} ∪ 𝑣)
3837uneq1i 3134 . . . . . . . 8 ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = (({𝑧} ∪ 𝑣) ∪ (𝐴𝑣))
39 unass 3141 . . . . . . . 8 (({𝑧} ∪ 𝑣) ∪ (𝐴𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣)))
4038, 39eqtri 2103 . . . . . . 7 ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣)))
4136, 40syl6reqr 2134 . . . . . 6 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = 𝐴)
42 unass 3141 . . . . . . . 8 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧})))
43 uncom 3128 . . . . . . . . . 10 ({𝑧} ∪ (𝐴 ∖ {𝑧})) = ((𝐴 ∖ {𝑧}) ∪ {𝑧})
44 simp1 939 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
4544ad3antrrr 476 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
46 dcdifsnid 6194 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑧𝐴) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
4745, 30, 46syl2anc 403 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
4843, 47syl5eq 2127 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ (𝐴 ∖ {𝑧})) = 𝐴)
4948uneq2d 3138 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧}))) = (𝑣𝐴))
5042, 49syl5eq 2127 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣𝐴))
51 simplrl 502 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑣𝐵)
5251, 27sstrd 3020 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑣𝐴)
53 ssequn1 3154 . . . . . . . 8 (𝑣𝐴 ↔ (𝑣𝐴) = 𝐴)
5452, 53sylib 120 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (𝑣𝐴) = 𝐴)
5550, 54eqtrd 2115 . . . . . 6 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = 𝐴)
5641, 55ineq12d 3186 . . . . 5 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧}))) = (𝐴𝐴))
5725, 56syl5eq 2127 . . . 4 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (𝐴𝐴))
58 inidm 3193 . . . 4 (𝐴𝐴) = 𝐴
5957, 58syl6req 2132 . . 3 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
6059ex 113 . 2 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) → (𝐴 = (𝑣 ∪ (𝐴𝑣)) → 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
61 simp2 940 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
624, 8, 12, 16, 21, 60, 61findcard2sd 6537 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  DECID wdc 776  w3a 920   = wceq 1285  wcel 1434  wral 2353  cdif 2981  cun 2982  cin 2983  wss 2984  c0 3269  {csn 3422  Fincfn 6386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 3999  ax-un 4223  ax-setind 4315  ax-iinf 4365
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-if 3374  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4083  df-iord 4156  df-on 4158  df-suc 4161  df-iom 4368  df-xp 4406  df-rel 4407  df-cnv 4408  df-co 4409  df-dm 4410  df-rn 4411  df-res 4412  df-ima 4413  df-iota 4933  df-fun 4970  df-fn 4971  df-f 4972  df-f1 4973  df-fo 4974  df-f1o 4975  df-fv 4976  df-er 6221  df-en 6387  df-fin 6389
This theorem is referenced by:  undiffi  6561
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