Theorem disjpreima 32606

Description: A preimage of a disjoint set is disjoint. (Contributed by Thierry Arnoux, 7-Feb-2017.)

Assertion

Ref	Expression
disjpreima	⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ 𝐴 𝐵) → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

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.

Step	Hyp	Ref	Expression
1		inpreima 7097	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)))
2		imaeq2 6085	. . . . . . . . . 10 ⊢ ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = (◡𝐹 “ ∅))
3		ima0 6106	. . . . . . . . . 10 ⊢ (◡𝐹 “ ∅) = ∅
4	2, 3	eqtrdi 2796	. . . . . . . . 9 ⊢ ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
5	1, 4	sylan9req 2801	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
6	5	ex 412	. . . . . . 7 ⊢ (Fun 𝐹 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅))
7		csbima12 6108	. . . . . . . . . 10 ⊢ ⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
8		csbconstg 3940	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌◡𝐹 = ◡𝐹)
9	8	elv 3493	. . . . . . . . . . 11 ⊢ ⦋𝑦 / 𝑥⦌◡𝐹 = ◡𝐹
10	9	imaeq1i 6086	. . . . . . . . . 10 ⊢ (⦋𝑦 / 𝑥⦌◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) = (◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
11	7, 10	eqtri 2768	. . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) = (◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
12		csbima12 6108	. . . . . . . . . 10 ⊢ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵) = (⦋𝑧 / 𝑥⦌◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
13		csbconstg 3940	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ V → ⦋𝑧 / 𝑥⦌◡𝐹 = ◡𝐹)
14	13	elv 3493	. . . . . . . . . . 11 ⊢ ⦋𝑧 / 𝑥⦌◡𝐹 = ◡𝐹
15	14	imaeq1i 6086	. . . . . . . . . 10 ⊢ (⦋𝑧 / 𝑥⦌◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵) = (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
16	12, 15	eqtri 2768	. . . . . . . . 9 ⊢ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵) = (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
17	11, 16	ineq12i 4239	. . . . . . . 8 ⊢ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵))
18	17	eqeq1i 2745	. . . . . . 7 ⊢ ((⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅ ↔ ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
19	6, 18	imbitrrdi 252	. . . . . 6 ⊢ (Fun 𝐹 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅))
20	19	orim2d 967	. . . . 5 ⊢ (Fun 𝐹 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅)))
21	20	ralimdv 3175	. . . 4 ⊢ (Fun 𝐹 → (∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅)))
22	21	ralimdv 3175	. . 3 ⊢ (Fun 𝐹 → (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅)))
23		disjors 5149	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
24		disjors 5149	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅))
25	22, 23, 24	3imtr4g 296	. 2 ⊢ (Fun 𝐹 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵)))
26	25	imp 406	1 ⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ 𝐴 𝐵) → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537  ∀wral 3067  Vcvv 3488  ⦋csb 3921   ∩ cin 3975  ∅c0 4352  Disj wdisj 5133  ^◡ccnv 5699   “ cima 5703  Fun wfun 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-disj 5134  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575

This theorem is referenced by:  fnpreimac  32689  elrspunidl  33421  sibfof  34305  dstrvprob  34436