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Theorem undifdc 6741
Description: Union of complementary parts into whole. This is a case where we can strengthen undifss 3390 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 3135 . . . 4 (𝑤 = ∅ → (𝐴𝑤) = (𝐴 ∖ ∅))
31, 2uneq12d 3178 . . 3 (𝑤 = ∅ → (𝑤 ∪ (𝐴𝑤)) = (∅ ∪ (𝐴 ∖ ∅)))
43eqeq2d 2111 . 2 (𝑤 = ∅ → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))))
5 id 19 . . . 4 (𝑤 = 𝑣𝑤 = 𝑣)
6 difeq2 3135 . . . 4 (𝑤 = 𝑣 → (𝐴𝑤) = (𝐴𝑣))
75, 6uneq12d 3178 . . 3 (𝑤 = 𝑣 → (𝑤 ∪ (𝐴𝑤)) = (𝑣 ∪ (𝐴𝑣)))
87eqeq2d 2111 . 2 (𝑤 = 𝑣 → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (𝑣 ∪ (𝐴𝑣))))
9 id 19 . . . 4 (𝑤 = (𝑣 ∪ {𝑧}) → 𝑤 = (𝑣 ∪ {𝑧}))
10 difeq2 3135 . . . 4 (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴𝑤) = (𝐴 ∖ (𝑣 ∪ {𝑧})))
119, 10uneq12d 3178 . . 3 (𝑤 = (𝑣 ∪ {𝑧}) → (𝑤 ∪ (𝐴𝑤)) = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
1211eqeq2d 2111 . 2 (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
13 id 19 . . . 4 (𝑤 = 𝐵𝑤 = 𝐵)
14 difeq2 3135 . . . 4 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
1513, 14uneq12d 3178 . . 3 (𝑤 = 𝐵 → (𝑤 ∪ (𝐴𝑤)) = (𝐵 ∪ (𝐴𝐵)))
1615eqeq2d 2111 . 2 (𝑤 = 𝐵 → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (𝐵 ∪ (𝐴𝐵))))
17 un0 3343 . . . 4 ((𝐴 ∖ ∅) ∪ ∅) = (𝐴 ∖ ∅)
18 uncom 3167 . . . 4 ((𝐴 ∖ ∅) ∪ ∅) = (∅ ∪ (𝐴 ∖ ∅))
19 dif0 3380 . . . 4 (𝐴 ∖ ∅) = 𝐴
2017, 18, 193eqtr3ri 2129 . . 3 𝐴 = (∅ ∪ (𝐴 ∖ ∅))
2120a1i 9 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (∅ ∪ (𝐴 ∖ ∅)))
22 difundi 3275 . . . . . . 7 (𝐴 ∖ (𝑣 ∪ {𝑧})) = ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧}))
2322uneq2i 3174 . . . . . 6 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = ((𝑣 ∪ {𝑧}) ∪ ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧})))
24 undi 3271 . . . . . 6 ((𝑣 ∪ {𝑧}) ∪ ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
2523, 24eqtri 2120 . . . . 5 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
26 simp3 951 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐵𝐴)
2726ad3antrrr 479 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐵𝐴)
28 simplrr 506 . . . . . . . . . . . 12 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧 ∈ (𝐵𝑣))
2928eldifad 3032 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧𝐵)
3027, 29sseldd 3048 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧𝐴)
3130snssd 3612 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → {𝑧} ⊆ 𝐴)
32 ssequn1 3193 . . . . . . . . 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 3177 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ 𝐴) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣))))
3633, 35eqtr3d 2134 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣))))
37 uncom 3167 . . . . . . . . 9 (𝑣 ∪ {𝑧}) = ({𝑧} ∪ 𝑣)
3837uneq1i 3173 . . . . . . . 8 ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = (({𝑧} ∪ 𝑣) ∪ (𝐴𝑣))
39 unass 3180 . . . . . . . 8 (({𝑧} ∪ 𝑣) ∪ (𝐴𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣)))
4038, 39eqtri 2120 . . . . . . 7 ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣)))
4136, 40syl6reqr 2151 . . . . . 6 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = 𝐴)
42 unass 3180 . . . . . . . 8 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧})))
43 uncom 3167 . . . . . . . . . 10 ({𝑧} ∪ (𝐴 ∖ {𝑧})) = ((𝐴 ∖ {𝑧}) ∪ {𝑧})
44 simp1 949 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
4544ad3antrrr 479 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
46 dcdifsnid 6330 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑧𝐴) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
4745, 30, 46syl2anc 406 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
4843, 47syl5eq 2144 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ (𝐴 ∖ {𝑧})) = 𝐴)
4948uneq2d 3177 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧}))) = (𝑣𝐴))
5042, 49syl5eq 2144 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣𝐴))
51 simplrl 505 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑣𝐵)
5251, 27sstrd 3057 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑣𝐴)
53 ssequn1 3193 . . . . . . . 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 2132 . . . . . 6 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = 𝐴)
5641, 55ineq12d 3225 . . . . 5 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧}))) = (𝐴𝐴))
5725, 56syl5eq 2144 . . . 4 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (𝐴𝐴))
58 inidm 3232 . . . 4 (𝐴𝐴) = 𝐴
5957, 58syl6req 2149 . . 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 950 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
624, 8, 12, 16, 21, 60, 61findcard2sd 6715 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  DECID wdc 786  w3a 930   = wceq 1299  wcel 1448  wral 2375  cdif 3018  cun 3019  cin 3020  wss 3021  c0 3310  {csn 3474  Fincfn 6564
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 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-iinf 4440
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-int 3719  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-tr 3967  df-id 4153  df-iord 4226  df-on 4228  df-suc 4231  df-iom 4443  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-er 6359  df-en 6565  df-fin 6567
This theorem is referenced by:  undiffi  6742
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