Theorem domssl 8983

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 489	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐵 ≼ 𝐶)
2		reldom 8937	. . . 4 ⊢ Rel ≼
3	2	brrelex12i 5707	. . 3 ⊢ (𝐵 ≼ 𝐶 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4		simpl 487	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ⊆ 𝐵)
5		ssexg 5284	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
6	5	adantrr 729	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ∈ V)
7		simprr 784	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐶 ∈ V)
8	4, 6, 7	jca32 524	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
9	3, 8	sylan2 604	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → (𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
10		brdomi 8944	. . 3 ⊢ (𝐵 ≼ 𝐶 → ∃𝑓 𝑓:𝐵–1-1→𝐶)
11		f1ssres 6773	. . . . . 6 ⊢ ((𝑓:𝐵–1-1→𝐶 ∧ 𝐴 ⊆ 𝐵) → (𝑓 ↾ 𝐴):𝐴–1-1→𝐶)
12		vex 3461	. . . . . . . . 9 ⊢ 𝑓 ∈ V
13	12	resex 6019	. . . . . . . 8 ⊢ (𝑓 ↾ 𝐴) ∈ V
14		f1dom4g 8950	. . . . . . . 8 ⊢ ((((𝑓 ↾ 𝐴) ∈ V ∧ 𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓 ↾ 𝐴):𝐴–1-1→𝐶) → 𝐴 ≼ 𝐶)
15	13, 14	mp3anl1 1479	. . . . . . 7 ⊢ (((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓 ↾ 𝐴):𝐴–1-1→𝐶) → 𝐴 ≼ 𝐶)
16	15	ancoms 463	. . . . . 6 ⊢ (((𝑓 ↾ 𝐴):𝐴–1-1→𝐶 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶)
17	11, 16	sylan 591	. . . . 5 ⊢ (((𝑓:𝐵–1-1→𝐶 ∧ 𝐴 ⊆ 𝐵) ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶)
18	17	expl 462	. . . 4 ⊢ (𝑓:𝐵–1-1→𝐶 → ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶))
19	18	exlimiv 1953	. . 3 ⊢ (∃𝑓 𝑓:𝐵–1-1→𝐶 → ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶))
20	10, 19	syl 18	. 2 ⊢ (𝐵 ≼ 𝐶 → ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ≼ 𝐶))
21	1, 9, 20	sylc 66	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → 𝐴 ≼ 𝐶)

Syntax hints:   → wi 4   ∧ wa 400  ∃wex 1802   ∈ wcel 2145  Vcvv 3457   ⊆ wss 3907   class class class wbr 5105   ↾ cres 5654  –1-1→wf1 6522   ≼ cdom 8929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-dom 8933

This theorem is referenced by:  ssct  9034  1sdom2dom  9202