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Theorem invdisj 4670
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 nfra2 2975 . . 3 𝑦𝑥𝐴𝑦𝐵 𝐶 = 𝑥
2 df-ral 2946 . . . . 5 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥))
3 rsp 2958 . . . . . . . . 9 (∀𝑦𝐵 𝐶 = 𝑥 → (𝑦𝐵𝐶 = 𝑥))
4 eqcom 2658 . . . . . . . . 9 (𝐶 = 𝑥𝑥 = 𝐶)
53, 4syl6ib 241 . . . . . . . 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 1779 . . . . 5 (∀𝑥(𝑥𝐴 → ∀𝑦𝐵 𝐶 = 𝑥) → ∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
92, 8sylbi 207 . . . 4 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶))
10 mo2icl 3418 . . . 4 (∀𝑥((𝑥𝐴𝑦𝐵) → 𝑥 = 𝐶) → ∃*𝑥(𝑥𝐴𝑦𝐵))
119, 10syl 17 . . 3 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∃*𝑥(𝑥𝐴𝑦𝐵))
121, 11alrimi 2120 . 2 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥 → ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
13 dfdisj2 4654 . 2 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦𝐵))
1412, 13sylibr 224 1 (∀𝑥𝐴𝑦𝐵 𝐶 = 𝑥Disj 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521   = wceq 1523  wcel 2030  ∃*wmo 2499  wral 2941  Disj wdisj 4652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rmo 2949  df-v 3233  df-disj 4653
This theorem is referenced by:  invdisjrab  4671  ackbijnn  14604  incexc2  14614  phisum  15542  itg1addlem1  23504  musum  24962  lgsquadlem1  25150  lgsquadlem2  25151  disjabrex  29521  disjabrexf  29522  actfunsnrndisj  30811  poimirlem27  33566
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