Theorem disjpreima 32966

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 6063	. . . . . . . . . 10 ⊢ ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = (◡𝐹 “ ∅))
3		ima0 6084	. . . . . . . . . 10 ⊢ (◡𝐹 “ ∅) = ∅
4	2, 3	eqtrdi 2817	. . . . . . . . 9 ⊢ ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
5	1, 4	sylan9req 2822	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
6	5	ex 418	. . . . . . 7 ⊢ (Fun 𝐹 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅))
7		csbima12 6086	. . . . . . . . . 10 ⊢ ⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
8		csbconstg 3875	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌◡𝐹 = ◡𝐹)
9	8	elv 3463	. . . . . . . . . . 11 ⊢ ⦋𝑦 / 𝑥⦌◡𝐹 = ◡𝐹
10	9	imaeq1i 6064	. . . . . . . . . 10 ⊢ (⦋𝑦 / 𝑥⦌◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) = (◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
11	7, 10	eqtri 2789	. . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) = (◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
12		csbima12 6086	. . . . . . . . . 10 ⊢ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵) = (⦋𝑧 / 𝑥⦌◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
13		csbconstg 3875	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ V → ⦋𝑧 / 𝑥⦌◡𝐹 = ◡𝐹)
14	13	elv 3463	. . . . . . . . . . 11 ⊢ ⦋𝑧 / 𝑥⦌◡𝐹 = ◡𝐹
15	14	imaeq1i 6064	. . . . . . . . . 10 ⊢ (⦋𝑧 / 𝑥⦌◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵) = (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
16	12, 15	eqtri 2789	. . . . . . . . 9 ⊢ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵) = (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
17	11, 16	ineq12i 4174	. . . . . . . 8 ⊢ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵))
18	17	eqeq1i 2771	. . . . . . 7 ⊢ ((⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅ ↔ ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
19	6, 18	imbitrrdi 255	. . . . . 6 ⊢ (Fun 𝐹 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅))
20	19	orim2d 982	. . . . 5 ⊢ (Fun 𝐹 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅)))
21	20	ralimdv 3182	. . . 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 3182	. . 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 5097	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
24		disjors 5097	. . 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 299	. 2 ⊢ (Fun 𝐹 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵)))
26	25	imp 412	1 ⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ 𝐴 𝐵) → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570  ∀wral 3082  Vcvv 3458  ⦋csb 3856   ∩ cin 3907  ∅c0 4289  Disj wdisj 5081  ^◡ccnv 5665   “ cima 5669  Fun wfun 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rmo 3372  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-disj 5082  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-fun 6545

This theorem is used by:  fnpreimac  33052  elrspunidl  33767  sibfof  34762  dstrvprob  34894