Theorem sess2 4268

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 3166	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V))
2		rabss2 3185	. . . . 5 ⊢ (𝐴 ⊆ 𝐵 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥})
3		ssexg 4075	. . . . . 6 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∧ {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
4	3	ex 114	. . . . 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 2503	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
7	1, 6	syld 45	. 2 ⊢ (𝐴 ⊆ 𝐵 → (∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
8		df-se 4263	. 2 ⊢ (𝑅 Se 𝐵 ↔ ∀𝑥 ∈ 𝐵 {𝑦 ∈ 𝐵 ∣ 𝑦𝑅𝑥} ∈ V)
9		df-se 4263	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
10	7, 8, 9	3imtr4g 204	1 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴))

Syntax hints:   → wi 4   ∈ wcel 1481  ∀wral 2417  {crab 2421  Vcvv 2689   ⊆ wss 3076   class class class wbr 3937   Se wse 4259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-in 3082  df-ss 3089  df-se 4263

This theorem is referenced by:  seeq2  4270