Theorem sess2 4060

Description: Subset theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.)

Assertion

Ref	Expression
sess2	⊢ (A ⊆ B → (𝑅 Se B → 𝑅 Se A))

Proof of Theorem sess2

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssralv 2998	. . 3 ⊢ (A ⊆ B → (∀x ∈ B {y ∈ B ∣ y𝑅x} ∈ V → ∀x ∈ A {y ∈ B ∣ y𝑅x} ∈ V))
2		rabss2 3017	. . . . 5 ⊢ (A ⊆ B → {y ∈ A ∣ y𝑅x} ⊆ {y ∈ B ∣ y𝑅x})
3		ssexg 3887	. . . . . 6 ⊢ (({y ∈ A ∣ y𝑅x} ⊆ {y ∈ B ∣ y𝑅x} ∧ {y ∈ B ∣ y𝑅x} ∈ V) → {y ∈ A ∣ y𝑅x} ∈ V)
4	3	ex 108	. . . . 5 ⊢ ({y ∈ A ∣ y𝑅x} ⊆ {y ∈ B ∣ y𝑅x} → ({y ∈ B ∣ y𝑅x} ∈ V → {y ∈ A ∣ y𝑅x} ∈ V))
5	2, 4	syl 14	. . . 4 ⊢ (A ⊆ B → ({y ∈ B ∣ y𝑅x} ∈ V → {y ∈ A ∣ y𝑅x} ∈ V))
6	5	ralimdv 2382	. . 3 ⊢ (A ⊆ B → (∀x ∈ A {y ∈ B ∣ y𝑅x} ∈ V → ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V))
7	1, 6	syld 40	. 2 ⊢ (A ⊆ B → (∀x ∈ B {y ∈ B ∣ y𝑅x} ∈ V → ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V))
8		df-se 4056	. 2 ⊢ (𝑅 Se B ↔ ∀x ∈ B {y ∈ B ∣ y𝑅x} ∈ V)
9		df-se 4056	. 2 ⊢ (𝑅 Se A ↔ ∀x ∈ A {y ∈ A ∣ y𝑅x} ∈ V)
10	7, 8, 9	3imtr4g 194	1 ⊢ (A ⊆ B → (𝑅 Se B → 𝑅 Se A))

Syntax hints:   → wi 4   ∈ wcel 1390  ∀wral 2300  {crab 2304  Vcvv 2551   ⊆ wss 2911   class class class wbr 3755   Se wse 4055

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-in 2918  df-ss 2925  df-se 4056

This theorem is referenced by:  seeq2  4062