Theorem domssl 9034

Description: If 𝐴 is a subset of 𝐵 and 𝐶 dominates 𝐵, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.)

Assertion

Ref	Expression
domssl	⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)

Proof of Theorem domssl

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐵 ≼ 𝐶)
2		reldom 8987	. . . 4 ⊢ Rel ≼
3	2	brrelex12i 5738	. . 3 ⊢ (𝐵 ≼ 𝐶 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4		simpl 482	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ⊆ 𝐵)
5		ssexg 5321	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
6	5	adantrr 717	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ∈ V)
7		simprr 773	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐶 ∈ V)
8	4, 6, 7	jca32 515	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
9	3, 8	sylan2 593	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → (𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
10		brdomi 8995	. . 3 ⊢ (𝐵 ≼ 𝐶 → ∃𝑓 𝑓:𝐵–1-1→𝐶)
11		f1ssres 6809	. . . . . 6 ⊢ ((𝑓:𝐵–1-1→𝐶 ∧ 𝐴 ⊆ 𝐵) → (𝑓 ↾ 𝐴):𝐴–1-1→𝐶)
12		vex 3483	. . . . . . . . 9 ⊢ 𝑓 ∈ V
13	12	resex 6045	. . . . . . . 8 ⊢ (𝑓 ↾ 𝐴) ∈ V
14		f1dom4g 9002	. . . . . . . 8 ⊢ ((((𝑓 ↾ 𝐴) ∈ V ∧ 𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓 ↾ 𝐴):𝐴–1-1→𝐶) → 𝐴 ≼ 𝐶)
15	13, 14	mp3anl1 1457	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓 ↾ 𝐴):𝐴–1-1→𝐶) → 𝐴 ≼ 𝐶)
16	15	ancoms 458	. . . . . 6 ⊢ (((𝑓 ↾ 𝐴):𝐴–1-1→𝐶 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶)
17	11, 16	sylan 580	. . . . 5 ⊢ (((𝑓:𝐵–1-1→𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶)
18	17	expl 457	. . . 4 ⊢ (𝑓:𝐵–1-1→𝐶 → ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶))
19	18	exlimiv 1930	. . 3 ⊢ (∃𝑓 𝑓:𝐵–1-1→𝐶 → ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶))
20	10, 19	syl 17	. 2 ⊢ (𝐵 ≼ 𝐶 → ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶))
21	1, 9, 20	sylc 65	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2108  Vcvv 3479   ⊆ wss 3950   class class class wbr 5141   ↾ cres 5685  –1-1→wf1 6556   ≼ cdom 8979

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-dom 8983

This theorem is referenced by:  ssct  9087  1sdom2dom  9279