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Theorem invdisj 5129
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 3299 . . 3 𝑦𝑥𝐴𝑦𝐵 𝐶 = 𝑥
2 df-ral 3062 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥))
3 rsp 3247 . . . . . . . . 9 (∀𝑦𝐵 𝐶 = 𝑥 → (𝑦𝐵𝐶 = 𝑥))
4 eqcom 2744 . . . . . . . . 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 1811 . . . . 5 (∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
92, 8sylbi 217 . . . 4 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
10 mo2icl 3720 . . . 4 (∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶) → ∃*𝑥(𝑥𝐴𝑦𝐵))
119, 10syl 17 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∃*𝑥(𝑥𝐴𝑦𝐵))
121, 11alrimi 2213 . 2 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
13 dfdisj2 5112 . 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 1538   = wceq 1540  wcel 2108  ∃*wmo 2538  wral 3061  Disj wdisj 5110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rmo 3380  df-v 3482  df-disj 5111
This theorem is referenced by:  invdisjrab  5130  ackbijnn  15864  incexc2  15874  phisum  16828  itg1addlem1  25727  musum  27234  lgsquadlem1  27424  lgsquadlem2  27425  disjabrex  32595  disjabrexf  32596  actfunsnrndisj  34620  poimirlem27  37654
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