Theorem elpredgg 6161

Description: Membership in a predecessor class. (Contributed by Scott Fenton, 17-Apr-2011.) Generalize to closed form. (Revised by BJ, 16-Oct-2024.)

Assertion

Ref	Expression
elpredgg	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))

Proof of Theorem elpredgg

Step	Hyp	Ref	Expression
1		df-pred 6149	. . 3 ⊢ Pred(𝑅, 𝐴, 𝑋) = (𝐴 ∩ (◡𝑅 “ {𝑋}))
2	1	elin2 4101	. 2 ⊢ (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})))
3		elinisegg 5950	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ (◡𝑅 “ {𝑋}) ↔ 𝑌𝑅𝑋))
4	3	anbi2d 632	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → ((𝑌 ∈ 𝐴 ∧ 𝑌 ∈ (◡𝑅 “ {𝑋})) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))
5	2, 4	syl5bb 286	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → (𝑌 ∈ Pred(𝑅, 𝐴, 𝑋) ↔ (𝑌 ∈ 𝐴 ∧ 𝑌𝑅𝑋)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∈ wcel 2110  {csn 4531   class class class wbr 5043  ^◡ccnv 5539   “ cima 5543  Predcpred 6148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-br 5044  df-opab 5106  df-xp 5546  df-cnv 5548  df-dm 5550  df-rn 5551  df-res 5552  df-ima 5553  df-pred 6149

This theorem is referenced by:  elpredg  6162  elpredimg  6163  elpred  6165