Theorem predep 6332

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 6303	. 2 ⊢ Pred( E , 𝐴, 𝑋) = (𝐴 ∩ (◡ E “ {𝑋}))
2		relcnv 6100	. . . . 5 ⊢ Rel ◡ E
3		relimasn 6083	. . . . 5 ⊢ (Rel ◡ E → (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦})
4	2, 3	ax-mp 5	. . . 4 ⊢ (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦}
5		brcnvg 5857	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋))
6	5	elvd 3457	. . . . . 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 2895	. . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑦 ∣ 𝑋◡ E 𝑦} = 𝑋)
10	4, 9	eqtrid 2808	. . 3 ⊢ (𝑋 ∈ 𝐵 → (◡ E “ {𝑋}) = 𝑋)
11	10	ineq2d 4166	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝐴 ∩ (◡ E “ {𝑋})) = (𝐴 ∩ 𝑋))
12	1, 11	eqtrid 2808	1 ⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  {cab 2739  Vcvv 3451   ∩ cin 3898  {csn 4584   class class class wbr 5103   E cep 5550  ^◡ccnv 5650   “ cima 5654  Rel wrel 5656  Predcpred 6302

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  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 6303

This theorem is used by:  trpred  6333