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Theorem disjpreima 30458
 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 6830 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)))
2 imaeq2 5902 . . . . . . . . . 10 ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = (𝐹 “ ∅))
3 ima0 5922 . . . . . . . . . 10 (𝐹 “ ∅) = ∅
42, 3eqtrdi 2809 . . . . . . . . 9 ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = ∅)
51, 4sylan9req 2814 . . . . . . . 8 ((Fun 𝐹 ∧ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅)
65ex 416 . . . . . . 7 (Fun 𝐹 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅))
7 csbima12 5924 . . . . . . . . . 10 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
8 csbconstg 3826 . . . . . . . . . . . 12 (𝑦 ∈ V → 𝑦 / 𝑥𝐹 = 𝐹)
98elv 3415 . . . . . . . . . . 11 𝑦 / 𝑥𝐹 = 𝐹
109imaeq1i 5903 . . . . . . . . . 10 (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐹𝑦 / 𝑥𝐵)
117, 10eqtri 2781 . . . . . . . . 9 𝑦 / 𝑥(𝐹𝐵) = (𝐹𝑦 / 𝑥𝐵)
12 csbima12 5924 . . . . . . . . . 10 𝑧 / 𝑥(𝐹𝐵) = (𝑧 / 𝑥𝐹𝑧 / 𝑥𝐵)
13 csbconstg 3826 . . . . . . . . . . . 12 (𝑧 ∈ V → 𝑧 / 𝑥𝐹 = 𝐹)
1413elv 3415 . . . . . . . . . . 11 𝑧 / 𝑥𝐹 = 𝐹
1514imaeq1i 5903 . . . . . . . . . 10 (𝑧 / 𝑥𝐹𝑧 / 𝑥𝐵) = (𝐹𝑧 / 𝑥𝐵)
1612, 15eqtri 2781 . . . . . . . . 9 𝑧 / 𝑥(𝐹𝐵) = (𝐹𝑧 / 𝑥𝐵)
1711, 16ineq12i 4117 . . . . . . . 8 (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵))
1817eqeq1i 2763 . . . . . . 7 ((𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅ ↔ ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅)
196, 18syl6ibr 255 . . . . . 6 (Fun 𝐹 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅))
2019orim2d 964 . . . . 5 (Fun 𝐹 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
2120ralimdv 3109 . . . 4 (Fun 𝐹 → (∀𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ∀𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
2221ralimdv 3109 . . 3 (Fun 𝐹 → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
23 disjors 5017 . . 3 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
24 disjors 5017 . . 3 (Disj 𝑥𝐴 (𝐹𝐵) ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅))
2522, 23, 243imtr4g 299 . 2 (Fun 𝐹 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 (𝐹𝐵)))
2625imp 410 1 ((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 844   = wceq 1538  ∀wral 3070  Vcvv 3409  ⦋csb 3807   ∩ cin 3859  ∅c0 4227  Disj wdisj 5001  ◡ccnv 5527   “ cima 5531  Fun wfun 6334 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-disj 5002  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-fun 6342 This theorem is referenced by:  fnpreimac  30544  elrspunidl  31139  sibfof  31838  dstrvprob  31969
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