Theorem dfpred4 39548

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 6319	. 2 ⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = {𝑚 ∈ 𝐴 ∣ 𝑚𝑅𝑁})
2		ec1cnvres 39345	. 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 1569   ∈ wcel 2144  {crab 3412   class class class wbr 5102  ^◡ccnv 5650   ↾ cres 5653  Predcpred 6306  [cec 8723

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307  df-ec 8727

This theorem is used by:  dfpre4  39549