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Theorem invdisj 3729
Description: If there is a function 𝐶(y) such that 𝐶(y) = x for all y B(x), then the sets B(x) for distinct x A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
invdisj (x A y B 𝐶 = xDisj x A B)
Distinct variable groups:   x,y   y,A   y,B   x,𝐶
Allowed substitution hints:   A(x)   B(x)   𝐶(y)

Proof of Theorem invdisj
StepHypRef Expression
1 nfra2xy 2338 . . 3 yx A y B 𝐶 = x
2 df-ral 2285 . . . . 5 (x A y B 𝐶 = xx(x Ay B 𝐶 = x))
3 rsp 2343 . . . . . . . . 9 (y B 𝐶 = x → (y B𝐶 = x))
4 eqcom 2020 . . . . . . . . 9 (𝐶 = xx = 𝐶)
53, 4syl6ib 150 . . . . . . . 8 (y B 𝐶 = x → (y Bx = 𝐶))
65imim2i 12 . . . . . . 7 ((x Ay B 𝐶 = x) → (x A → (y Bx = 𝐶)))
76impd 242 . . . . . 6 ((x Ay B 𝐶 = x) → ((x A y B) → x = 𝐶))
87alimi 1320 . . . . 5 (x(x Ay B 𝐶 = x) → x((x A y B) → x = 𝐶))
92, 8sylbi 114 . . . 4 (x A y B 𝐶 = xx((x A y B) → x = 𝐶))
10 mo2icl 2693 . . . 4 (x((x A y B) → x = 𝐶) → ∃*x(x A y B))
119, 10syl 14 . . 3 (x A y B 𝐶 = x∃*x(x A y B))
121, 11alrimi 1392 . 2 (x A y B 𝐶 = xy∃*x(x A y B))
13 dfdisj2 3717 . 2 (Disj x A By∃*x(x A y B))
1412, 13sylibr 137 1 (x A y B 𝐶 = xDisj x A B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1224   = wceq 1226   wcel 1370  ∃*wmo 1879  wral 2280  Disj wdisj 3715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rmo 2288  df-v 2533  df-disj 3716
This theorem is referenced by: (None)
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