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Theorem disjpreima 31815
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 7066 . . . . . . . . 9 (Fun 𝐹 → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)))
2 imaeq2 6056 . . . . . . . . . 10 ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = (𝐹 “ ∅))
3 ima0 6077 . . . . . . . . . 10 (𝐹 “ ∅) = ∅
42, 3eqtrdi 2789 . . . . . . . . 9 ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝐹 “ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵)) = ∅)
51, 4sylan9req 2794 . . . . . . . 8 ((Fun 𝐹 ∧ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅)
65ex 414 . . . . . . 7 (Fun 𝐹 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅))
7 csbima12 6079 . . . . . . . . . 10 𝑦 / 𝑥(𝐹𝐵) = (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵)
8 csbconstg 3913 . . . . . . . . . . . 12 (𝑦 ∈ V → 𝑦 / 𝑥𝐹 = 𝐹)
98elv 3481 . . . . . . . . . . 11 𝑦 / 𝑥𝐹 = 𝐹
109imaeq1i 6057 . . . . . . . . . 10 (𝑦 / 𝑥𝐹𝑦 / 𝑥𝐵) = (𝐹𝑦 / 𝑥𝐵)
117, 10eqtri 2761 . . . . . . . . 9 𝑦 / 𝑥(𝐹𝐵) = (𝐹𝑦 / 𝑥𝐵)
12 csbima12 6079 . . . . . . . . . 10 𝑧 / 𝑥(𝐹𝐵) = (𝑧 / 𝑥𝐹𝑧 / 𝑥𝐵)
13 csbconstg 3913 . . . . . . . . . . . 12 (𝑧 ∈ V → 𝑧 / 𝑥𝐹 = 𝐹)
1413elv 3481 . . . . . . . . . . 11 𝑧 / 𝑥𝐹 = 𝐹
1514imaeq1i 6057 . . . . . . . . . 10 (𝑧 / 𝑥𝐹𝑧 / 𝑥𝐵) = (𝐹𝑧 / 𝑥𝐵)
1612, 15eqtri 2761 . . . . . . . . 9 𝑧 / 𝑥(𝐹𝐵) = (𝐹𝑧 / 𝑥𝐵)
1711, 16ineq12i 4211 . . . . . . . 8 (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵))
1817eqeq1i 2738 . . . . . . 7 ((𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅ ↔ ((𝐹𝑦 / 𝑥𝐵) ∩ (𝐹𝑧 / 𝑥𝐵)) = ∅)
196, 18syl6ibr 252 . . . . . 6 (Fun 𝐹 → ((𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅ → (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅))
2019orim2d 966 . . . . 5 (Fun 𝐹 → ((𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
2120ralimdv 3170 . . . 4 (Fun 𝐹 → (∀𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ∀𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
2221ralimdv 3170 . . 3 (Fun 𝐹 → (∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅) → ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅)))
23 disjors 5130 . . 3 (Disj 𝑥𝐴 𝐵 ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥𝐵𝑧 / 𝑥𝐵) = ∅))
24 disjors 5130 . . 3 (Disj 𝑥𝐴 (𝐹𝐵) ↔ ∀𝑦𝐴𝑧𝐴 (𝑦 = 𝑧 ∨ (𝑦 / 𝑥(𝐹𝐵) ∩ 𝑧 / 𝑥(𝐹𝐵)) = ∅))
2522, 23, 243imtr4g 296 . 2 (Fun 𝐹 → (Disj 𝑥𝐴 𝐵Disj 𝑥𝐴 (𝐹𝐵)))
2625imp 408 1 ((Fun 𝐹Disj 𝑥𝐴 𝐵) → Disj 𝑥𝐴 (𝐹𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wral 3062  Vcvv 3475  csb 3894  cin 3948  c0 4323  Disj wdisj 5114  ccnv 5676  cima 5680  Fun wfun 6538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rmo 3377  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-disj 5115  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546
This theorem is referenced by:  fnpreimac  31896  elrspunidl  32546  sibfof  33339  dstrvprob  33470
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