Theorem indval0 12196

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 12194	. . . . . 6 ⊢ (𝑂 ∈ V → (𝟭‘𝑂) = (𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0))))
2	1	fveq1d 6865	. . . . 5 ⊢ (𝑂 ∈ V → ((𝟭‘𝑂)‘𝐴) = ((𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))‘𝐴))
3	2	adantr 484	. . . 4 ⊢ ((𝑂 ∈ V ∧ ¬ 𝐴 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐴) = ((𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))‘𝐴))
4		elpwi 4561	. . . . . . 7 ⊢ (𝐴 ∈ 𝒫 𝑂 → 𝐴 ⊆ 𝑂)
5	4	con3i 154	. . . . . 6 ⊢ (¬ 𝐴 ⊆ 𝑂 → ¬ 𝐴 ∈ 𝒫 𝑂)
6	5	adantl 485	. . . . 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 7003	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝒫 𝑂 → ((𝑎 ∈ 𝒫 𝑂 ↦ (𝑥 ∈ 𝑂 ↦ if(𝑥 ∈ 𝑎, 1, 0)))‘𝐴) = ∅)
9	6, 8	syl 17	. . . 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 416	. 2 ⊢ (𝑂 ∈ V → (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅))
12		fv2prc 6905	. . 3 ⊢ (¬ 𝑂 ∈ V → ((𝟭‘𝑂)‘𝐴) = ∅)
13	12	a1d 25	. 2 ⊢ (¬ 𝑂 ∈ V → (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅))
14	11, 13	pm2.61i 183	1 ⊢ (¬ 𝐴 ⊆ 𝑂 → ((𝟭‘𝑂)‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  Vcvv 3453   ⊆ wss 3904  ∅c0 4285  ifcif 4479  𝒫 cpw 4554   ↦ cmpt 5180  ‘cfv 6517  0cc0 11070  1c1 11071  𝟭cind 12192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-ind 12193

This theorem is referenced by: (None)