Theorem suppvalbrOLD 8209

Description: Obsolete version of suppvalbr 8208 as of 14-Oct-2026. The value of the operation constructing the support of a function expressed by binary relations. (Contributed by AV, 7-Apr-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
suppvalbrOLD	⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))})

Distinct variable groups:   𝑥,𝑅,𝑦   𝑥,𝑍,𝑦

Allowed substitution hints:   𝑉(𝑥, 𝑦)   𝑊(𝑥, 𝑦)

Proof of Theorem suppvalbrOLD

Step	Hyp	Ref	Expression
1		df-rab 3417	. . . 4 ⊢ {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}} = {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})}
2		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
3	2	eldm 5889	. . . . . 6 ⊢ (𝑥 ∈ dom 𝑅 ↔ ∃𝑦 𝑥𝑅𝑦)
4		imasng 6094	. . . . . . . . 9 ⊢ (𝑥 ∈ V → (𝑅 “ {𝑥}) = {𝑦 ∣ 𝑥𝑅𝑦})
5	4	elv 3459	. . . . . . . 8 ⊢ (𝑅 “ {𝑥}) = {𝑦 ∣ 𝑥𝑅𝑦}
6	5	neeq1i 3023	. . . . . . 7 ⊢ ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ {𝑦 ∣ 𝑥𝑅𝑦} ≠ {𝑍})
7		df-sn 4589	. . . . . . . 8 ⊢ {𝑍} = {𝑦 ∣ 𝑦 = 𝑍}
8	7	neeq2i 3024	. . . . . . 7 ⊢ ({𝑦 ∣ 𝑥𝑅𝑦} ≠ {𝑍} ↔ {𝑦 ∣ 𝑥𝑅𝑦} ≠ {𝑦 ∣ 𝑦 = 𝑍})
9		nabbib 3064	. . . . . . 7 ⊢ ({𝑦 ∣ 𝑥𝑅𝑦} ≠ {𝑦 ∣ 𝑦 = 𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
10	6, 8, 9	3bitri 300	. . . . . 6 ⊢ ((𝑅 “ {𝑥}) ≠ {𝑍} ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
11	3, 10	anbi12i 640	. . . . 5 ⊢ ((𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍}) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
12	11	abbii 2830	. . . 4 ⊢ {𝑥 ∣ (𝑥 ∈ dom 𝑅 ∧ (𝑅 “ {𝑥}) ≠ {𝑍})} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
13	1, 12	eqtr2i 2787	. . 3 ⊢ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}}
14	13	a1i 11	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))} = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}})
15		df-ne 2960	. . . . . . 7 ⊢ (𝑦 ≠ 𝑍 ↔ ¬ 𝑦 = 𝑍)
16	15	bibi2i 340	. . . . . 6 ⊢ ((𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍) ↔ (𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
17	16	exbii 1881	. . . . 5 ⊢ (∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍) ↔ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))
18	17	anbi2i 635	. . . 4 ⊢ ((∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍)) ↔ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍)))
19	18	abbii 2830	. . 3 ⊢ {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))}
20	19	a1i 11	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))} = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ ¬ 𝑦 = 𝑍))})
21		suppval 8206	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∈ dom 𝑅 ∣ (𝑅 “ {𝑥}) ≠ {𝑍}})
22	14, 20, 21	3eqtr4rd 2809	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝑍 ∈ 𝑊) → (𝑅 supp 𝑍) = {𝑥 ∣ (∃𝑦 𝑥𝑅𝑦 ∧ ∃𝑦(𝑥𝑅𝑦 ↔ 𝑦 ≠ 𝑍))})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  {crab 3416  Vcvv 3454  {csn 4588   class class class wbr 5106  dom cdm 5655   “ cima 5658  (class class class)co 7443   supp csupp 8204

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-supp 8205

This theorem is used by: (None)