Theorem sess1 4074

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

Assertion

Ref	Expression
sess1	⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴))

Proof of Theorem sess1

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

Step	Hyp	Ref	Expression
1		simpl 102	. . . . . 6 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → 𝑅 ⊆ 𝑆)
2	1	ssbrd 3805	. . . . 5 ⊢ ((𝑅 ⊆ 𝑆 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 → 𝑦𝑆𝑥))
3	2	ss2rabdv 3021	. . . 4 ⊢ (𝑅 ⊆ 𝑆 → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥})
4		ssexg 3896	. . . . 5 ⊢ (({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∧ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V) → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
5	4	ex 108	. . . 4 ⊢ ({𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ⊆ {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
6	3, 5	syl 14	. . 3 ⊢ (𝑅 ⊆ 𝑆 → ({𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
7	6	ralimdv 2388	. 2 ⊢ (𝑅 ⊆ 𝑆 → (∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V))
8		df-se 4070	. 2 ⊢ (𝑆 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑆𝑥} ∈ V)
9		df-se 4070	. 2 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐴 ∣ 𝑦𝑅𝑥} ∈ V)
10	7, 8, 9	3imtr4g 194	1 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴))

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1393  ∀wral 2306  {crab 2310  Vcvv 2557   ⊆ wss 2917   class class class wbr 3764   Se wse 4066

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-br 3765  df-se 4070

This theorem is referenced by:  seeq1  4076