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Mirrors > Home > MPE Home > Th. List > domssr | Structured version Visualization version GIF version |
Description: If 𝐶 is a superset of 𝐵 and 𝐵 dominates 𝐴, then 𝐶 also dominates 𝐴. (Contributed by BTernaryTau, 7-Dec-2024.) |
Ref | Expression |
---|---|
domssr | ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≼ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 8895 | . . 3 ⊢ (𝐴 ≼ 𝐵 → ∃𝑓 𝑓:𝐴–1-1→𝐵) | |
2 | 1 | 3ad2ant3 1135 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → ∃𝑓 𝑓:𝐴–1-1→𝐵) |
3 | simp2 1137 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐵 ⊆ 𝐶) | |
4 | reldom 8886 | . . . . 5 ⊢ Rel ≼ | |
5 | 4 | brrelex1i 5687 | . . . 4 ⊢ (𝐴 ≼ 𝐵 → 𝐴 ∈ V) |
6 | 5 | 3ad2ant3 1135 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐴 ∈ V) |
7 | simp1 1136 | . . 3 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐶 ∈ 𝑉) | |
8 | 3, 6, 7 | jca32 516 | . 2 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → (𝐵 ⊆ 𝐶 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ 𝑉))) |
9 | f1ss 6742 | . . . . 5 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝑓:𝐴–1-1→𝐶) | |
10 | vex 3448 | . . . . . . 7 ⊢ 𝑓 ∈ V | |
11 | f1dom4g 8902 | . . . . . . 7 ⊢ (((𝑓 ∈ V ∧ 𝐴 ∈ V ∧ 𝐶 ∈ 𝑉) ∧ 𝑓:𝐴–1-1→𝐶) → 𝐴 ≼ 𝐶) | |
12 | 10, 11 | mp3anl1 1455 | . . . . . 6 ⊢ (((𝐴 ∈ V ∧ 𝐶 ∈ 𝑉) ∧ 𝑓:𝐴–1-1→𝐶) → 𝐴 ≼ 𝐶) |
13 | 12 | ancoms 459 | . . . . 5 ⊢ ((𝑓:𝐴–1-1→𝐶 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ 𝑉)) → 𝐴 ≼ 𝐶) |
14 | 9, 13 | sylan 580 | . . . 4 ⊢ (((𝑓:𝐴–1-1→𝐵 ∧ 𝐵 ⊆ 𝐶) ∧ (𝐴 ∈ V ∧ 𝐶 ∈ 𝑉)) → 𝐴 ≼ 𝐶) |
15 | 14 | expl 458 | . . 3 ⊢ (𝑓:𝐴–1-1→𝐵 → ((𝐵 ⊆ 𝐶 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ 𝑉)) → 𝐴 ≼ 𝐶)) |
16 | 15 | exlimiv 1933 | . 2 ⊢ (∃𝑓 𝑓:𝐴–1-1→𝐵 → ((𝐵 ⊆ 𝐶 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ 𝑉)) → 𝐴 ≼ 𝐶)) |
17 | 2, 8, 16 | sylc 65 | 1 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐵 ⊆ 𝐶 ∧ 𝐴 ≼ 𝐵) → 𝐴 ≼ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 ∃wex 1781 ∈ wcel 2106 Vcvv 3444 ⊆ wss 3909 class class class wbr 5104 –1-1→wf1 6491 ≼ cdom 8878 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-dom 8882 |
This theorem is referenced by: 0sdom1dom 9179 rex2dom 9187 |
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