Theorem dfpred4 38800

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 6277	. 2 ⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = {𝑚 ∈ 𝐴 ∣ 𝑚𝑅𝑁})
2		ec1cnvres 38597	. 2 ⊢ (𝑁 ∈ 𝑉 → [𝑁]◡(𝑅 ↾ 𝐴) = {𝑚 ∈ 𝐴 ∣ 𝑚𝑅𝑁})
3	1, 2	eqtr4d 2774	1 ⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = [𝑁]◡(𝑅 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3389   class class class wbr 5085  ^◡ccnv 5630   ↾ cres 5633  Predcpred 6264  [cec 8641

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ec 8645

This theorem is referenced by:  dfpre4  38801