Theorem dfse2 5499

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 5074	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
2		dfrab3 3902	. . . . 5 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥})
3		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
4		iniseg 5496	. . . . . . 7 ⊢ (𝑥 ∈ V → (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥})
5	3, 4	ax-mp 5	. . . . . 6 ⊢ (◡𝑅 “ {𝑥}) = {𝑦 ∣ 𝑦𝑅𝑥}
6	5	ineq2i 3811	. . . . 5 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑥})) = (𝐴 ∩ {𝑦 ∣ 𝑦𝑅𝑥})
7	2, 6	eqtr4i 2647	. . . 4 ⊢ {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} = (𝐴 ∩ (◡𝑅 “ {𝑥}))
8	7	eleq1i 2692	. . 3 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)
9	8	ralbii 2980	. 2 ⊢ (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)
10	1, 9	bitri 264	1 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 (𝐴 ∩ (◡𝑅 “ {𝑥})) ∈ V)

Syntax hints:   ↔ wb 196   = wceq 1483   ∈ wcel 1990  {cab 2608  ∀wral 2912  {crab 2916  Vcvv 3200   ∩ cin 3573  {csn 4177   class class class wbr 4653   Se wse 5071  ^◡ccnv 5113   “ cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-se 5074  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  isoselem  6591  fnse  7294