Theorem dfproplem 38555

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 6990 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 4953	. . . . 5 ⊢ ∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 𝑥 ∈ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)})}
2		df-sn 4585	. . . . . . . . . 10 ⊢ {(prop¬‘𝑧)} = {𝑥 ∣ 𝑥 = (prop¬‘𝑧)}
3		iunsn 5024	. . . . . . . . . 10 ⊢ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)} = {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)}
4	2, 3	uneq12i 4113	. . . . . . . . 9 ⊢ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) = ({𝑥 ∣ 𝑥 = (prop¬‘𝑧)} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)})
5		unab 4254	. . . . . . . . 9 ⊢ ({𝑥 ∣ 𝑥 = (prop¬‘𝑧)} ∪ {𝑥 ∣ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)}) = {𝑥 ∣ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))}
6	4, 5	eqtri 2783	. . . . . . . 8 ⊢ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) = {𝑥 ∣ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))}
7	6	eqabri 2902	. . . . . . 7 ⊢ (𝑥 ∈ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ↔ (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)))
8	7	rexbii 3109	. . . . . 6 ⊢ (∃𝑧 ∈ 𝐴 𝑥 ∈ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ↔ ∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧)))
9	8	abbii 2827	. . . . 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 2783	. . . 4 ⊢ ∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) = {𝑥 ∣ ∃𝑧 ∈ 𝐴 (𝑥 = (prop¬‘𝑧) ∨ ∃𝑤 ∈ 𝐴 𝑥 = (𝑤prop→𝑧))}
11		iunsn 5024	. . . 4 ⊢ ∪ 𝑛 ∈ ℕ {(propvar ‘𝑛)} = {𝑥 ∣ ∃𝑛 ∈ ℕ 𝑥 = (propvar ‘𝑛)}
12	10, 11	uneq12i 4113	. . 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 4254	. . 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 2783	. 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 5397	. . . . 5 ⊢ {(prop¬‘𝑧)} ∈ V
17		snex 5397	. . . . . 6 ⊢ {(𝑤prop→𝑧)} ∈ V
18	15, 17	iunex 7964	. . . . 5 ⊢ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)} ∈ V
19	16, 18	unex 7745	. . . 4 ⊢ ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ∈ V
20	15, 19	iunex 7964	. . 3 ⊢ ∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ∈ V
21		nnex 12296	. . . 4 ⊢ ℕ ∈ V
22		snex 5397	. . . 4 ⊢ {(propvar ‘𝑛)} ∈ V
23	21, 22	iunex 7964	. . 3 ⊢ ∪ 𝑛 ∈ ℕ {(propvar ‘𝑛)} ∈ V
24	20, 23	unex 7745	. 2 ⊢ (∪ 𝑧 ∈ 𝐴 ({(prop¬‘𝑧)} ∪ ∪ 𝑤 ∈ 𝐴 {(𝑤prop→𝑧)}) ∪ ∪ 𝑛 ∈ ℕ {(propvar ‘𝑛)}) ∈ V
25	14, 24	eqeltrri 2857	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 2145  {cab 2738  ∃wrex 3086  Vcvv 3450   ∪ cun 3897  {csn 4584  ∪ ciun 4951  ‘cfv 6528  (class class class)co 7409  ℕcn 12290  propvar cpropvar 38547  prop¬cpropneg 38548  prop→cpropimp 38549

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 2732  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735  ax-cnex 11213  ax-1cn 11215  ax-addcl 11217

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  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-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6294  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-f 6532  df-f1 6533  df-fo 6534  df-f1o 6535  df-fv 6536  df-ov 7412  df-om 7862  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-nn 12291

This theorem is used by:  varprop  38556  negprop  38557  impprop  38558  dfprop2  38560