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Theorem disjpreima 28779
 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 6250 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)))
2 imaeq2 5381 . . . . . . . . . 10 ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = (𝐹 “ ∅))
3 ima0 5400 . . . . . . . . . 10 (𝐹 “ ∅) = ∅
42, 3syl6eq 2660 . . . . . . . . 9 ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = ∅)
51, 4sylan9req 2665 . . . . . . . 8 ((Fun 𝐹 ∧ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅)
65ex 449 . . . . . . 7 (Fun 𝐹 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅))
7 csbima12 5402 . . . . . . . . . 10 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
8 vex 3176 . . . . . . . . . . . 12 𝑦 ∈ V
9 csbconstg 3512 . . . . . . . . . . . 12 (𝑦 ∈ V → 𝑦 / 𝑥𝐹 = 𝐹)
108, 9ax-mp 5 . . . . . . . . . . 11 𝑦 / 𝑥𝐹 = 𝐹
1110imaeq1i 5382 . . . . . . . . . 10 (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐹𝑦 / 𝑥𝐵)
127, 11eqtri 2632 . . . . . . . . 9 𝑦 / 𝑥(𝐹𝐵) = (𝐹𝑦 / 𝑥𝐵)
13 csbima12 5402 . . . . . . . . . 10 𝑧 / 𝑥(𝐹𝐵) = (𝑧 / 𝑥𝐹𝑧 / 𝑥𝐵)
14 vex 3176 . . . . . . . . . . . 12 𝑧 ∈ V
15 csbconstg 3512 . . . . . . . . . . . 12 (𝑧 ∈ V → 𝑧 / 𝑥𝐹 = 𝐹)
1614, 15ax-mp 5 . . . . . . . . . . 11 𝑧 / 𝑥𝐹 = 𝐹
1716imaeq1i 5382 . . . . . . . . . 10 (𝑧 / 𝑥𝐹𝑧 / 𝑥𝐵) = (𝐹𝑧 / 𝑥𝐵)
1813, 17eqtri 2632 . . . . . . . . 9 𝑧 / 𝑥(𝐹𝐵) = (𝐹𝑧 / 𝑥𝐵)
1912, 18ineq12i 3774 . . . . . . . 8 (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵))
2019eqeq1i 2615 . . . . . . 7 ((𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅ ↔ ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅)
216, 20syl6ibr 241 . . . . . 6 (Fun 𝐹 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅))
2221orim2d 881 . . . . 5 (Fun 𝐹 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
2322ralimdv 2946 . . . 4 (Fun 𝐹 → (∀𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ∀𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
2423ralimdv 2946 . . 3 (Fun 𝐹 → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
25 disjors 4568 . . 3 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
26 disjors 4568 . . 3 (Disj 𝑥𝐴 (𝐹𝐵) ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅))
2724, 25, 263imtr4g 284 . 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 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  ⦋csb 3499   ∩ cin 3539  ∅c0 3874  Disj wdisj 4553  ◡ccnv 5037   “ cima 5041  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-disj 4554  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-fun 5806 This theorem is referenced by:  sibfof  29729  dstrvprob  29860
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