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Theorem undifdc 6953
Description: Union of complementary parts into whole. This is a case where we can strengthen undifss 3518 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 3262 . . . 4 (𝑤 = ∅ → (𝐴𝑤) = (𝐴 ∖ ∅))
31, 2uneq12d 3305 . . 3 (𝑤 = ∅ → (𝑤 ∪ (𝐴𝑤)) = (∅ ∪ (𝐴 ∖ ∅)))
43eqeq2d 2201 . 2 (𝑤 = ∅ → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (∅ ∪ (𝐴 ∖ ∅))))
5 id 19 . . . 4 (𝑤 = 𝑣𝑤 = 𝑣)
6 difeq2 3262 . . . 4 (𝑤 = 𝑣 → (𝐴𝑤) = (𝐴𝑣))
75, 6uneq12d 3305 . . 3 (𝑤 = 𝑣 → (𝑤 ∪ (𝐴𝑤)) = (𝑣 ∪ (𝐴𝑣)))
87eqeq2d 2201 . 2 (𝑤 = 𝑣 → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (𝑣 ∪ (𝐴𝑣))))
9 id 19 . . . 4 (𝑤 = (𝑣 ∪ {𝑧}) → 𝑤 = (𝑣 ∪ {𝑧}))
10 difeq2 3262 . . . 4 (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴𝑤) = (𝐴 ∖ (𝑣 ∪ {𝑧})))
119, 10uneq12d 3305 . . 3 (𝑤 = (𝑣 ∪ {𝑧}) → (𝑤 ∪ (𝐴𝑤)) = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
1211eqeq2d 2201 . 2 (𝑤 = (𝑣 ∪ {𝑧}) → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
13 id 19 . . . 4 (𝑤 = 𝐵𝑤 = 𝐵)
14 difeq2 3262 . . . 4 (𝑤 = 𝐵 → (𝐴𝑤) = (𝐴𝐵))
1513, 14uneq12d 3305 . . 3 (𝑤 = 𝐵 → (𝑤 ∪ (𝐴𝑤)) = (𝐵 ∪ (𝐴𝐵)))
1615eqeq2d 2201 . 2 (𝑤 = 𝐵 → (𝐴 = (𝑤 ∪ (𝐴𝑤)) ↔ 𝐴 = (𝐵 ∪ (𝐴𝐵))))
17 un0 3471 . . . 4 ((𝐴 ∖ ∅) ∪ ∅) = (𝐴 ∖ ∅)
18 uncom 3294 . . . 4 ((𝐴 ∖ ∅) ∪ ∅) = (∅ ∪ (𝐴 ∖ ∅))
19 dif0 3508 . . . 4 (𝐴 ∖ ∅) = 𝐴
2017, 18, 193eqtr3ri 2219 . . 3 𝐴 = (∅ ∪ (𝐴 ∖ ∅))
2120a1i 9 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (∅ ∪ (𝐴 ∖ ∅)))
22 difundi 3402 . . . . . . 7 (𝐴 ∖ (𝑣 ∪ {𝑧})) = ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧}))
2322uneq2i 3301 . . . . . 6 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = ((𝑣 ∪ {𝑧}) ∪ ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧})))
24 undi 3398 . . . . . 6 ((𝑣 ∪ {𝑧}) ∪ ((𝐴𝑣) ∩ (𝐴 ∖ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
2523, 24eqtri 2210 . . . . 5 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})))
26 uncom 3294 . . . . . . . . 9 (𝑣 ∪ {𝑧}) = ({𝑧} ∪ 𝑣)
2726uneq1i 3300 . . . . . . . 8 ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = (({𝑧} ∪ 𝑣) ∪ (𝐴𝑣))
28 unass 3307 . . . . . . . 8 (({𝑧} ∪ 𝑣) ∪ (𝐴𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣)))
2927, 28eqtri 2210 . . . . . . 7 ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣)))
30 simp3 1001 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐵𝐴)
3130ad3antrrr 492 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐵𝐴)
32 simplrr 536 . . . . . . . . . . . 12 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧 ∈ (𝐵𝑣))
3332eldifad 3155 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧𝐵)
3431, 33sseldd 3171 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑧𝐴)
3534snssd 3752 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → {𝑧} ⊆ 𝐴)
36 ssequn1 3320 . . . . . . . . 9 ({𝑧} ⊆ 𝐴 ↔ ({𝑧} ∪ 𝐴) = 𝐴)
3735, 36sylib 122 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ 𝐴) = 𝐴)
38 simpr 110 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = (𝑣 ∪ (𝐴𝑣)))
3938uneq2d 3304 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ 𝐴) = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣))))
4037, 39eqtr3d 2224 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = ({𝑧} ∪ (𝑣 ∪ (𝐴𝑣))))
4129, 40eqtr4id 2241 . . . . . 6 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) = 𝐴)
42 unass 3307 . . . . . . . 8 ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧})))
43 uncom 3294 . . . . . . . . . 10 ({𝑧} ∪ (𝐴 ∖ {𝑧})) = ((𝐴 ∖ {𝑧}) ∪ {𝑧})
44 simp1 999 . . . . . . . . . . . 12 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
4544ad3antrrr 492 . . . . . . . . . . 11 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
46 dcdifsnid 6530 . . . . . . . . . . 11 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑧𝐴) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
4745, 34, 46syl2anc 411 . . . . . . . . . 10 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝐴 ∖ {𝑧}) ∪ {𝑧}) = 𝐴)
4843, 47eqtrid 2234 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ({𝑧} ∪ (𝐴 ∖ {𝑧})) = 𝐴)
4948uneq2d 3304 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (𝑣 ∪ ({𝑧} ∪ (𝐴 ∖ {𝑧}))) = (𝑣𝐴))
5042, 49eqtrid 2234 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = (𝑣𝐴))
51 simplrl 535 . . . . . . . . 9 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑣𝐵)
5251, 31sstrd 3180 . . . . . . . 8 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝑣𝐴)
53 ssequn1 3320 . . . . . . . 8 (𝑣𝐴 ↔ (𝑣𝐴) = 𝐴)
5452, 53sylib 122 . . . . . . 7 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (𝑣𝐴) = 𝐴)
5550, 54eqtrd 2222 . . . . . 6 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧})) = 𝐴)
5641, 55ineq12d 3352 . . . . 5 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → (((𝑣 ∪ {𝑧}) ∪ (𝐴𝑣)) ∩ ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ {𝑧}))) = (𝐴𝐴))
5725, 56eqtrid 2234 . . . 4 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))) = (𝐴𝐴))
58 inidm 3359 . . . 4 (𝐴𝐴) = 𝐴
5957, 58eqtr2di 2239 . . 3 (((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) ∧ 𝐴 = (𝑣 ∪ (𝐴𝑣))) → 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧}))))
6059ex 115 . 2 ((((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) ∧ 𝑣 ∈ Fin) ∧ (𝑣𝐵𝑧 ∈ (𝐵𝑣))) → (𝐴 = (𝑣 ∪ (𝐴𝑣)) → 𝐴 = ((𝑣 ∪ {𝑧}) ∪ (𝐴 ∖ (𝑣 ∪ {𝑧})))))
61 simp2 1000 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐵 ∈ Fin)
624, 8, 12, 16, 21, 60, 61findcard2sd 6921 1 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝐵 ∈ Fin ∧ 𝐵𝐴) → 𝐴 = (𝐵 ∪ (𝐴𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  DECID wdc 835  w3a 980   = wceq 1364  wcel 2160  wral 2468  cdif 3141  cun 3142  cin 3143  wss 3144  c0 3437  {csn 3607  Fincfn 6767
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-er 6560  df-en 6768  df-fin 6770
This theorem is referenced by:  undiffi  6954
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