Theorem predep 6336

Description: The predecessor under the membership relation is equivalent to an intersection. (Contributed by Scott Fenton, 27-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
predep	⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))

Proof of Theorem predep

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pred 6307	. 2 ⊢ Pred( E , 𝐴, 𝑋) = (𝐴 ∩ (◡ E “ {𝑋}))
2		relcnv 6102	. . . . 5 ⊢ Rel ◡ E
3		relimasn 6085	. . . . 5 ⊢ (Rel ◡ E → (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦})
4	2, 3	ax-mp 5	. . . 4 ⊢ (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦}
5		brcnvg 5858	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋))
6	5	elvd 3456	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋))
7		epelg 5552	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑦 E 𝑋 ↔ 𝑦 ∈ 𝑋))
8	6, 7	bitrd 282	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝑋◡ E 𝑦 ↔ 𝑦 ∈ 𝑋))
9	8	eqabcdv 2894	. . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑦 ∣ 𝑋◡ E 𝑦} = 𝑋)
10	4, 9	eqtrid 2807	. . 3 ⊢ (𝑋 ∈ 𝐵 → (◡ E “ {𝑋}) = 𝑋)
11	10	ineq2d 4116	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝐴 ∩ (◡ E “ {𝑋})) = (𝐴 ∩ 𝑋))
12	1, 11	eqtrid 2807	1 ⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ∈ wcel 2144  {cab 2738  Vcvv 3450   ∩ cin 4088  {csn 4584   class class class wbr 5102   E cep 5550  ^◡ccnv 5650   “ cima 5654  Rel wrel 5656  Predcpred 6306

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-ne 2956  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-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6307

This theorem is used by:  trpred  6337