Theorem dfproplem 38832

Description: Given a set A, the set of all variables encoded as natural numbers, negations of elements in A, and implications between elements of A forms a set. This lemma is used when using fvmptd 7021 on the defining function of PROP. (Contributed by Thomas van Maaren, 21-Aug-2026.)

Hypothesis

Ref	Expression
dfproplem.a	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
dfproplem	⊢ {𝑥 ∣ (∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ∈ V

Distinct variable groups:   𝑤,𝐴,𝑥   𝑧,𝐴,𝑥   𝑥,𝑛

Allowed substitution hint:   𝐴(𝑛)

Proof of Theorem dfproplem

Step	Hyp	Ref	Expression
1		df-iun 4956	. . . . 5 ⊢ ∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 𝑥 ∈ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)})}
2		df-sn 4589	. . . . . . . . . 10 ⊢ {(prop¬‘𝑧)} = {𝑥 ∣ 𝑥 = (prop¬‘𝑧)}
3		iunsn 5027	. . . . . . . . . 10 ⊢ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)} = {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)}
4	2, 3	uneq12i 4168	. . . . . . . . 9 ⊢ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) = ({𝑥 ∣ 𝑥 = (prop¬‘𝑧)} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)})
5		unab 4256	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝑥 = (prop¬‘𝑧)} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)}) = {𝑥 ∣ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))}
6	4, 5	eqtri 2786	. . . . . . . 8 ⊢ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) = {𝑥 ∣ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))}
7	6	eqabri 2906	. . . . . . 7 ⊢ (𝑥 ∈ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ↔ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)))
8	7	rexbii 3113	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 𝑥 ∈ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ↔ ∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)))
9	8	abbii 2830	. . . . 5 ⊢ {𝑥 ∣ ∃𝑧 ∈ 𝐴 𝑥 ∈ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)})} = {𝑥 ∣ ∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))}
10	1, 9	eqtri 2786	. . . 4 ⊢ ∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))}
11		iunsn 5027	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ {(propvar ‘𝑛)} = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)}
12	10, 11	uneq12i 4168	. . 3 ⊢ (∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ∪ ∪ 𝑛 ∈ ℕ {(propvar ‘𝑛)}) = ({𝑥 ∣ ∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))} ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)})
13		unab 4256	. . 3 ⊢ ({𝑥 ∣ ∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))} ∪ {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)}) = {𝑥 ∣ (∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}
14	12, 13	eqtri 2786	. 2 ⊢ (∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ∪ ∪ 𝑛 ∈ ℕ {(propvar ‘𝑛)}) = {𝑥 ∣ (∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))}
15		dfproplem.a	. . . 4 ⊢ 𝐴 ∈ V
16		snex 5401	. . . . 5 ⊢ {(prop¬‘𝑧)} ∈ V
17		snex 5401	. . . . . 6 ⊢ {(𝑤prop→𝑧)} ∈ V
18	15, 17	iunex 8001	. . . . 5 ⊢ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)} ∈ V
19	16, 18	unex 7782	. . . 4 ⊢ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ∈ V
20	15, 19	iunex 8001	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ∈ V
21		nnex 12365	. . . 4 ⊢ ℕ ∈ V
22		snex 5401	. . . 4 ⊢ {(propvar ‘𝑛)} ∈ V
23	21, 22	iunex 8001	. . 3 ⊢ ∪ 𝑛 ∈ ℕ {(propvar ‘𝑛)} ∈ V
24	20, 23	unex 7782	. 2 ⊢ (∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ∪ ∪ 𝑛 ∈ ℕ {(propvar ‘𝑛)}) ∈ V
25	14, 24	eqeltrri 2861	1 ⊢ {𝑥 ∣ (∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)) ∨ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛))} ∈ V

This proof depends on syntax axioms:   ∨ wo 861   = wceq 1570   ∈ wcel 2147  {cab 2741  ∃wrex 3090  Vcvv 3454   ∪ cun 4152  {csn 4588  ∪ ciun 4954  ‘cfv 6555  (class class class)co 7440  ℕcn 12359  propvar cpropvar 38824  prop¬cpropneg 38825  prop→cpropimp 38826

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-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-1cn 11282  ax-addcl 11284

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  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-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  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-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-om 7899  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-nn 12360

This theorem is used by:  varprop  38833  negprop  38834  impprop  38835  dfprop2  38837