Theorem dfpred4 39228

Description: Alternate definition of the predecessor class when 𝑁 is a set. (Contributed by Peter Mazsa, 26-Jan-2026.)

Assertion

Ref	Expression
dfpred4	⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = [𝑁]◡(𝑅 ↾ 𝐴))

Proof of Theorem dfpred4

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfpred3g 6311	. 2 ⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = {𝑚 ∈ 𝐴 ∣ 𝑚𝑅𝑁})
2		ec1cnvres 39025	. 2 ⊢ (𝑁 ∈ 𝑉 → [𝑁]◡(𝑅 ↾ 𝐴) = {𝑚 ∈ 𝐴 ∣ 𝑚𝑅𝑁})
3	1, 2	eqtr4d 2798	1 ⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = [𝑁]◡(𝑅 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {crab 3412   class class class wbr 5103  ^◡ccnv 5654   ↾ cres 5657  Predcpred 6298  [cec 8695

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-ext 2732  ax-sep 5251  ax-pr 5398

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-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ec 8699

This theorem is used by:  dfpre4  39229