Theorem predep 6296

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 6267	. 2 ⊢ Pred( E , 𝐴, 𝑋) = (𝐴 ∩ (◡ E “ {𝑋}))
2		relcnv 6071	. . . . 5 ⊢ Rel ◡ E
3		relimasn 6052	. . . . 5 ⊢ (Rel ◡ E → (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦})
4	2, 3	ax-mp 5	. . . 4 ⊢ (◡ E “ {𝑋}) = {𝑦 ∣ 𝑋◡ E 𝑦}
5		brcnvg 5836	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑦 ∈ V) → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋))
6	5	elvd 3448	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑋◡ E 𝑦 ↔ 𝑦 E 𝑋))
7		epelg 5533	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (𝑦 E 𝑋 ↔ 𝑦 ∈ 𝑋))
8	6, 7	bitrd 279	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → (𝑋◡ E 𝑦 ↔ 𝑦 ∈ 𝑋))
9	8	eqabcdv 2871	. . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑦 ∣ 𝑋◡ E 𝑦} = 𝑋)
10	4, 9	eqtrid 2784	. . 3 ⊢ (𝑋 ∈ 𝐵 → (◡ E “ {𝑋}) = 𝑋)
11	10	ineq2d 4174	. 2 ⊢ (𝑋 ∈ 𝐵 → (𝐴 ∩ (◡ E “ {𝑋})) = (𝐴 ∩ 𝑋))
12	1, 11	eqtrid 2784	1 ⊢ (𝑋 ∈ 𝐵 → Pred( E , 𝐴, 𝑋) = (𝐴 ∩ 𝑋))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {cab 2715  Vcvv 3442   ∩ cin 3902  {csn 4582   class class class wbr 5100   E cep 5531  ^◡ccnv 5631   “ cima 5635  Rel wrel 5637  Predcpred 6266

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-ne 2934  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-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267

This theorem is referenced by:  trpred  6297