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Theorem invdisj 5134
Description: If there is a function 𝐶(𝑦) such that 𝐶(𝑦) = 𝑥 for all 𝑦𝐵(𝑥), then the sets 𝐵(𝑥) for distinct 𝑥𝐴 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 nfra2w 3297 . . 3 𝑦𝑥𝐴𝑦𝐵 𝐶 = 𝑥
2 df-ral 3060 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥))
3 rsp 3245 . . . . . . . . 9 (∀𝑦𝐵 𝐶 = 𝑥 → (𝑦𝐵𝐶 = 𝑥))
4 eqcom 2742 . . . . . . . . 9 (𝐶 = 𝑥𝑥 = 𝐶)
53, 4imbitrdi 251 . . . . . . . 8 (∀𝑦𝐵 𝐶 = 𝑥 → (𝑦𝐵𝑥 = 𝐶))
65imim2i 16 . . . . . . 7 ((𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥) → (𝑥𝐴 → (𝑦𝐵𝑥 = 𝐶)))
76impd 410 . . . . . 6 ((𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥) → ((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
87alimi 1808 . . . . 5 (∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
92, 8sylbi 217 . . . 4 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
10 mo2icl 3723 . . . 4 (∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶) → ∃*𝑥(𝑥𝐴𝑦𝐵))
119, 10syl 17 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∃*𝑥(𝑥𝐴𝑦𝐵))
121, 11alrimi 2211 . 2 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
13 dfdisj2 5117 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
1412, 13sylibr 234 1 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1535   = wceq 1537  wcel 2106  ∃*wmo 2536  wral 3059  Disj wdisj 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rmo 3378  df-v 3480  df-disj 5116
This theorem is referenced by:  invdisjrab  5135  ackbijnn  15861  incexc2  15871  phisum  16824  itg1addlem1  25741  musum  27249  lgsquadlem1  27439  lgsquadlem2  27440  disjabrex  32602  disjabrexf  32603  actfunsnrndisj  34599  poimirlem27  37634
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