Theorem dfse2 5140

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 4459	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
2		dfrab3 3501	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥})
3		vex 2818	. . . . . . 7 ⊢ 𝑥 ∈ V
4		iniseg 5139	. . . . . . 7 ⊢ (𝑥 ∈ V → (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥})
5	3, 4	ax-mp 5	. . . . . 6 ⊢ (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥}
6	5	ineq2i 3423	. . . . 5 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥})
7	2, 6	eqtr4i 2258	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ (◡𝑅 “ {𝑥}))
8	7	eleq1i 2300	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)
9	8	ralbii 2550	. 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)
10	1, 9	bitri 184	1 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)

Syntax hints:   ↔ wb 105   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  {crab 2526  Vcvv 2815   ∩ cin 3213  {csn 3694   class class class wbr 4114   Se wse 4455  ^◡ccnv 4753   “ cima 4757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-se 4459  df-xp 4760  df-cnv 4762  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767

This theorem is referenced by:  isoselem  5999