Theorem domssl 8939

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 8893	. . . 4 ⊢ Rel ≼
3	2	brrelex12i 5680	. . 3 ⊢ (𝐵 ≼ 𝐶 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4		simpl 482	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ⊆ 𝐵)
5		ssexg 5269	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V)
6	5	adantrr 718	. . . 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 594	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ≼ 𝐶) → (𝐴 ⊆ 𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
10		brdomi 8900	. . 3 ⊢ (𝐵 ≼ 𝐶 → ∃𝑓 𝑓:𝐵–1-1→𝐶)
11		f1ssres 6738	. . . . . 6 ⊢ ((𝑓:𝐵–1-1→𝐶 ∧ 𝐴 ⊆ 𝐵) → (𝑓 ↾ 𝐴):𝐴–1-1→𝐶)
12		vex 3445	. . . . . . . . 9 ⊢ 𝑓 ∈ V
13	12	resex 5989	. . . . . . . 8 ⊢ (𝑓 ↾ 𝐴) ∈ V
14		f1dom4g 8906	. . . . . . . 8 ⊢ ((((𝑓 ↾ 𝐴) ∈ V ∧ 𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓 ↾ 𝐴):𝐴–1-1→𝐶) → 𝐴 ≼ 𝐶)
15	13, 14	mp3anl1 1458	. . . . . . 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 581	. . . . 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 1932	. . 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 1781   ∈ wcel 2114  Vcvv 3441   ⊆ wss 3902   class class class wbr 5099   ↾ cres 5627  –1-1→wf1 6490   ≼ cdom 8885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-dom 8889

This theorem is referenced by:  ssct  8990  1sdom2dom  9158