Theorem indval0 12305

Description: The indicator function generator does not generate a (meaningful) indicator function for a class which is not a subset of the domain. (Contributed by AV, 11-Apr-2026.)

Assertion

Ref	Expression
indval0	⊢ (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅)

Proof of Theorem indval0

Dummy variables 𝑎 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		indv 12303	. . . . . 6 ⊢ (𝑂 ∈ V → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
2	1	fveq1d 6879	. . . . 5 ⊢ (𝑂 ∈ V → ((𝟭‘𝑂)‘𝐴) = ((𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))‘𝐴))
3	2	adantr 486	. . . 4 ⊢ ((𝑂 ∈ V ∧ ¬ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))‘𝐴))
4		elpwi 4564	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 𝑂 → 𝐴 ⊆ 𝑂)
5	4	con3i 155	. . . . . 6 ⊢ (¬ 𝐴 ⊆ 𝑂 → ¬ 𝐴 ∈ 𝒫 𝑂)
6	5	adantl 487	. . . . 5 ⊢ ((𝑂 ∈ V ∧ ¬ 𝐴 ⊆ 𝑂) → ¬ 𝐴 ∈ 𝒫 𝑂)
7		eqid 2761	. . . . . 6 ⊢ (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))
8	7	fvmptndm 7017	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝒫 𝑂 → ((𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))‘𝐴) = ∅)
9	6, 8	syl 18	. . . 4 ⊢ ((𝑂 ∈ V ∧ ¬ 𝐴 ⊆ 𝑂) → ((𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))‘𝐴) = ∅)
10	3, 9	eqtrd 2796	. . 3 ⊢ ((𝑂 ∈ V ∧ ¬ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ∅)
11	10	ex 418	. 2 ⊢ (𝑂 ∈ V → (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅))
12		fv2prc 6919	. . 3 ⊢ (¬ 𝑂 ∈ V → ((𝟭‘𝑂)‘𝐴) = ∅)
13	12	a1d 26	. 2 ⊢ (¬ 𝑂 ∈ V → (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅))
14	11, 13	pm2.61i 184	1 ⊢ (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  ifcif 4482  𝒫 cpw 4557   ↦ cmpt 5186  ‘cfv 6531  0cc0 11181  1c1 11182  𝟭cind 12301

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ind 12302

This theorem is used by: (None)