Theorem dfse2 6130

Description: Alternate definition of set-like relation. (Contributed by Mario Carneiro, 23-Jun-2015.)

Assertion

Ref	Expression
dfse2	⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑅

Proof of Theorem dfse2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-se 5653	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
2		dfrab3 4338	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥})
3		iniseg 6127	. . . . . . 7 ⊢ (𝑥 ∈ V → (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥})
4	3	elv 3493	. . . . . 6 ⊢ (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥}
5	4	ineq2i 4238	. . . . 5 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥})
6	2, 5	eqtr4i 2771	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ (◡𝑅 “ {𝑥}))
7	6	eleq1i 2835	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)
8	7	ralbii 3099	. 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)
9	1, 8	bitri 275	1 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  {crab 3443  Vcvv 3488   ∩ cin 3975  {csn 4648   class class class wbr 5166   Se wse 5650  ^◡ccnv 5699   “ cima 5703

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-se 5653  df-xp 5706  df-cnv 5708  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713

This theorem is referenced by:  dfse3  6368  isoselem  7377  fnse  8174