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Theorem invdisj 4571
 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 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 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 nfra2 2930 . . 3 𝑦𝑥𝐴𝑦𝐵 𝐶 = 𝑥
2 df-ral 2901 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥))
3 rsp 2913 . . . . . . . . 9 (∀𝑦𝐵 𝐶 = 𝑥 → (𝑦𝐵𝐶 = 𝑥))
4 eqcom 2617 . . . . . . . . 9 (𝐶 = 𝑥𝑥 = 𝐶)
53, 4syl6ib 240 . . . . . . . 8 (∀𝑦𝐵 𝐶 = 𝑥 → (𝑦𝐵𝑥 = 𝐶))
65imim2i 16 . . . . . . 7 ((𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥) → (𝑥𝐴 → (𝑦𝐵𝑥 = 𝐶)))
76impd 446 . . . . . 6 ((𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥) → ((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
87alimi 1730 . . . . 5 (∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
92, 8sylbi 206 . . . 4 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
10 mo2icl 3352 . . . 4 (∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶) → ∃*𝑥(𝑥𝐴𝑦𝐵))
119, 10syl 17 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∃*𝑥(𝑥𝐴𝑦𝐵))
121, 11alrimi 2069 . 2 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
13 dfdisj2 4555 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
1412, 13sylibr 223 1 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  ∀wral 2896  Disj wdisj 4553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rmo 2904  df-v 3175  df-disj 4554 This theorem is referenced by:  invdisjrab  4572  ackbijnn  14399  incexc2  14409  phisum  15333  itg1addlem1  23265  musum  24717  lgsquadlem1  24905  lgsquadlem2  24906  disjabrex  28777  disjabrexf  28778  poimirlem27  32606
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