Theorem disjpreima 32082

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 7064	. . . . . . . . 9 ⊢ (Fun 𝐹 → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)))
2		imaeq2 6054	. . . . . . . . . 10 ⊢ ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = (◡𝐹 “ ∅))
3		ima0 6075	. . . . . . . . . 10 ⊢ (◡𝐹 “ ∅) = ∅
4	2, 3	eqtrdi 2786	. . . . . . . . 9 ⊢ ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → (◡𝐹 “ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
5	1, 4	sylan9req 2791	. . . . . . . 8 ⊢ ((Fun 𝐹 ∧ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅)
6	5	ex 411	. . . . . . 7 ⊢ (Fun 𝐹 → ((⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅ → ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)) = ∅))
7		csbima12 6077	. . . . . . . . . 10 ⊢ ⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) = (⦋𝑦 / 𝑥⦌◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
8		csbconstg 3911	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ V → ⦋𝑦 / 𝑥⦌◡𝐹 = ◡𝐹)
9	8	elv 3478	. . . . . . . . . . 11 ⊢ ⦋𝑦 / 𝑥⦌◡𝐹 = ◡𝐹
10	9	imaeq1i 6055	. . . . . . . . . 10 ⊢ (⦋𝑦 / 𝑥⦌◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) = (◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
11	7, 10	eqtri 2758	. . . . . . . . 9 ⊢ ⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) = (◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵)
12		csbima12 6077	. . . . . . . . . 10 ⊢ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵) = (⦋𝑧 / 𝑥⦌◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
13		csbconstg 3911	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ V → ⦋𝑧 / 𝑥⦌◡𝐹 = ◡𝐹)
14	13	elv 3478	. . . . . . . . . . 11 ⊢ ⦋𝑧 / 𝑥⦌◡𝐹 = ◡𝐹
15	14	imaeq1i 6055	. . . . . . . . . 10 ⊢ (⦋𝑧 / 𝑥⦌◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵) = (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
16	12, 15	eqtri 2758	. . . . . . . . 9 ⊢ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵) = (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵)
17	11, 16	ineq12i 4209	. . . . . . . 8 ⊢ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ((◡𝐹 “ ⦋𝑦 / 𝑥⦌𝐵) ∩ (◡𝐹 “ ⦋𝑧 / 𝑥⦌𝐵))
18	17	eqeq1i 2735	. . . . . . 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 963	. . . . 5 ⊢ (Fun 𝐹 → ((𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅) → (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌(◡𝐹 “ 𝐵) ∩ ⦋𝑧 / 𝑥⦌(◡𝐹 “ 𝐵)) = ∅)))
21	20	ralimdv 3167	. . . 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 3167	. . 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 5128	. . 3 ⊢ (Disj 𝑥 ∈ 𝐴 𝐵 ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦 = 𝑧 ∨ (⦋𝑦 / 𝑥⦌𝐵 ∩ ⦋𝑧 / 𝑥⦌𝐵) = ∅))
24		disjors 5128	. . 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 295	. 2 ⊢ (Fun 𝐹 → (Disj 𝑥 ∈ 𝐴 𝐵 → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵)))
26	25	imp 405	1 ⊢ ((Fun 𝐹 ∧ Disj 𝑥 ∈ 𝐴 𝐵) → Disj 𝑥 ∈ 𝐴 (◡𝐹 “ 𝐵))

Syntax hints:   → wi 4   ∧ wa 394   ∨ wo 843   = wceq 1539  ∀wral 3059  Vcvv 3472  ⦋csb 3892   ∩ cin 3946  ∅c0 4321  Disj wdisj 5112  ^◡ccnv 5674   “ cima 5678  Fun wfun 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rmo 3374  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-disj 5113  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-fun 6544

This theorem is referenced by:  fnpreimac  32163  elrspunidl  32820  sibfof  33637  dstrvprob  33768