Theorem dfpred4 38730

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 6279	. 2 ⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = {𝑚 ∈ 𝐴 ∣ 𝑚𝑅𝑁})
2		ec1cnvres 38527	. 2 ⊢ (𝑁 ∈ 𝑉 → [𝑁]◡(𝑅 ↾ 𝐴) = {𝑚 ∈ 𝐴 ∣ 𝑚𝑅𝑁})
3	1, 2	eqtr4d 2775	1 ⊢ (𝑁 ∈ 𝑉 → Pred(𝑅, 𝐴, 𝑁) = [𝑁]◡(𝑅 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {crab 3401   class class class wbr 5100  ^◡ccnv 5631   ↾ cres 5634  Predcpred 6266  [cec 8643

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 2709  ax-sep 5243  ax-pr 5379

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 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ec 8647

This theorem is referenced by:  dfpre4  38731