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Theorem domssl 9058
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 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)

Proof of Theorem domssl
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . 2 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
2 reldom 9009 . . . 4 Rel ≼
32brrelex12i 5755 . . 3 (𝐵𝐶 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4 simpl 482 . . . 4 ((𝐴𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐵)
5 ssexg 5341 . . . . 5 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
65adantrr 716 . . . 4 ((𝐴𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ∈ V)
7 simprr 772 . . . 4 ((𝐴𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐶 ∈ V)
84, 6, 7jca32 515 . . 3 ((𝐴𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
93, 8sylan2 592 . 2 ((𝐴𝐵𝐵𝐶) → (𝐴𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
10 brdomi 9018 . . 3 (𝐵𝐶 → ∃𝑓 𝑓:𝐵1-1𝐶)
11 f1ssres 6824 . . . . . 6 ((𝑓:𝐵1-1𝐶𝐴𝐵) → (𝑓𝐴):𝐴1-1𝐶)
12 vex 3492 . . . . . . . . 9 𝑓 ∈ V
1312resex 6058 . . . . . . . 8 (𝑓𝐴) ∈ V
14 f1dom4g 9025 . . . . . . . 8 ((((𝑓𝐴) ∈ V ∧ 𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓𝐴):𝐴1-1𝐶) → 𝐴𝐶)
1513, 14mp3anl1 1455 . . . . . . 7 (((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓𝐴):𝐴1-1𝐶) → 𝐴𝐶)
1615ancoms 458 . . . . . 6 (((𝑓𝐴):𝐴1-1𝐶 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐶)
1711, 16sylan 579 . . . . 5 (((𝑓:𝐵1-1𝐶𝐴𝐵) ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐶)
1817expl 457 . . . 4 (𝑓:𝐵1-1𝐶 → ((𝐴𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐶))
1918exlimiv 1929 . . 3 (∃𝑓 𝑓:𝐵1-1𝐶 → ((𝐴𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐶))
2010, 19syl 17 . 2 (𝐵𝐶 → ((𝐴𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐶))
211, 9, 20sylc 65 1 ((𝐴𝐵𝐵𝐶) → 𝐴𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wex 1777  wcel 2108  Vcvv 3488  wss 3976   class class class wbr 5166  cres 5702  1-1wf1 6570  cdom 9001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-dom 9005
This theorem is referenced by:  ssct  9117  1sdom2dom  9310
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