Theorem disjpreima 31815

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 7066	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)))
2		imaeq2 6056	. . . . . . . . . 10 ⊢ ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = (◡𝐹 “ ∅))
3		ima0 6077	. . . . . . . . . 10 ⊢ (◡𝐹 “ ∅) = ∅
4	2, 3	eqtrdi 2789	. . . . . . . . 9 ⊢ ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
5	1, 4	sylan9req 2794	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
6	5	ex 414	. . . . . . 7 ⊢ (Fun 𝐹 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅))
7		csbima12 6079	. . . . . . . . . 10 ⊢ ⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
8		csbconstg 3913	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌◡𝐹 = ◡𝐹)
9	8	elv 3481	. . . . . . . . . . 11 ⊢ ⦋𝑦 / 𝑥⦌◡𝐹 = ◡𝐹
10	9	imaeq1i 6057	. . . . . . . . . 10 ⊢ (⦋𝑦 / 𝑥⦌◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) = (◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
11	7, 10	eqtri 2761	. . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) = (◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
12		csbima12 6079	. . . . . . . . . 10 ⊢ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵) = (⦋𝑧 / 𝑥⦌◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
13		csbconstg 3913	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ V → ⦋𝑧 / 𝑥⦌◡𝐹 = ◡𝐹)
14	13	elv 3481	. . . . . . . . . . 11 ⊢ ⦋𝑧 / 𝑥⦌◡𝐹 = ◡𝐹
15	14	imaeq1i 6057	. . . . . . . . . 10 ⊢ (⦋𝑧 / 𝑥⦌◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵) = (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
16	12, 15	eqtri 2761	. . . . . . . . 9 ⊢ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵) = (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
17	11, 16	ineq12i 4211	. . . . . . . 8 ⊢ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵))
18	17	eqeq1i 2738	. . . . . . 7 ⊢ ((⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅ ↔ ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
19	6, 18	imbitrrdi 251	. . . . . 6 ⊢ (Fun 𝐹 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅))
20	19	orim2d 966	. . . . 5 ⊢ (Fun 𝐹 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅)))
21	20	ralimdv 3170	. . . 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 3170	. . 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 5130	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
24		disjors 5130	. . 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 408	1 ⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ 𝐴 𝐵) → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

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