Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjpreima Structured version   Visualization version   GIF version

Theorem disjpreima 29523
Description: A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)
Assertion
Ref Expression
disjpreima ((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem disjpreima
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inpreima 6382 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)))
2 imaeq2 5497 . . . . . . . . . 10 ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = (𝐹 “ ∅))
3 ima0 5516 . . . . . . . . . 10 (𝐹 “ ∅) = ∅
42, 3syl6eq 2701 . . . . . . . . 9 ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = ∅)
51, 4sylan9req 2706 . . . . . . . 8 ((Fun 𝐹 ∧ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅)
65ex 449 . . . . . . 7 (Fun 𝐹 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅))
7 csbima12 5518 . . . . . . . . . 10 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
8 vex 3234 . . . . . . . . . . . 12 𝑦 ∈ V
9 csbconstg 3579 . . . . . . . . . . . 12 (𝑦 ∈ V → 𝑦 / 𝑥𝐹 = 𝐹)
108, 9ax-mp 5 . . . . . . . . . . 11 𝑦 / 𝑥𝐹 = 𝐹
1110imaeq1i 5498 . . . . . . . . . 10 (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐹𝑦 / 𝑥𝐵)
127, 11eqtri 2673 . . . . . . . . 9 𝑦 / 𝑥(𝐹𝐵) = (𝐹𝑦 / 𝑥𝐵)
13 csbima12 5518 . . . . . . . . . 10 𝑧 / 𝑥(𝐹𝐵) = (𝑧 / 𝑥𝐹𝑧 / 𝑥𝐵)
14 vex 3234 . . . . . . . . . . . 12 𝑧 ∈ V
15 csbconstg 3579 . . . . . . . . . . . 12 (𝑧 ∈ V → 𝑧 / 𝑥𝐹 = 𝐹)
1614, 15ax-mp 5 . . . . . . . . . . 11 𝑧 / 𝑥𝐹 = 𝐹
1716imaeq1i 5498 . . . . . . . . . 10 (𝑧 / 𝑥𝐹𝑧 / 𝑥𝐵) = (𝐹𝑧 / 𝑥𝐵)
1813, 17eqtri 2673 . . . . . . . . 9 𝑧 / 𝑥(𝐹𝐵) = (𝐹𝑧 / 𝑥𝐵)
1912, 18ineq12i 3845 . . . . . . . 8 (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵))
2019eqeq1i 2656 . . . . . . 7 ((𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅ ↔ ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅)
216, 20syl6ibr 242 . . . . . 6 (Fun 𝐹 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅))
2221orim2d 903 . . . . 5 (Fun 𝐹 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
2322ralimdv 2992 . . . 4 (Fun 𝐹 → (∀𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ∀𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
2423ralimdv 2992 . . 3 (Fun 𝐹 → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
25 disjors 4667 . . 3 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
26 disjors 4667 . . 3 (Disj 𝑥𝐴 (𝐹𝐵) ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅))
2724, 25, 263imtr4g 285 . 2 (Fun 𝐹 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 (𝐹𝐵)))
2827imp 444 1 ((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  csb 3566  cin 3606  c0 3948  Disj wdisj 4652  ccnv 5142  cima 5146  Fun wfun 5920
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  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  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-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-disj 4653  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-fun 5928
This theorem is referenced by:  sibfof  30530  dstrvprob  30661
  Copyright terms: Public domain W3C validator