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Theorem domssl 8938
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 485 . 2 ((𝐴𝐵𝐵𝐶) → 𝐵𝐶)
2 reldom 8889 . . . 4 Rel ≼
32brrelex12i 5687 . . 3 (𝐵𝐶 → (𝐵 ∈ V ∧ 𝐶 ∈ V))
4 simpl 483 . . . 4 ((𝐴𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐵)
5 ssexg 5280 . . . . 5 ((𝐴𝐵𝐵 ∈ V) → 𝐴 ∈ V)
65adantrr 715 . . . 4 ((𝐴𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐴 ∈ V)
7 simprr 771 . . . 4 ((𝐴𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → 𝐶 ∈ V)
84, 6, 7jca32 516 . . 3 ((𝐴𝐵 ∧ (𝐵 ∈ V ∧ 𝐶 ∈ V)) → (𝐴𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
93, 8sylan2 593 . 2 ((𝐴𝐵𝐵𝐶) → (𝐴𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)))
10 brdomi 8898 . . 3 (𝐵𝐶 → ∃𝑓 𝑓:𝐵1-1𝐶)
11 f1ssres 6746 . . . . . 6 ((𝑓:𝐵1-1𝐶𝐴𝐵) → (𝑓𝐴):𝐴1-1𝐶)
12 vex 3449 . . . . . . . . 9 𝑓 ∈ V
1312resex 5985 . . . . . . . 8 (𝑓𝐴) ∈ V
14 f1dom4g 8905 . . . . . . . 8 ((((𝑓𝐴) ∈ V ∧ 𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓𝐴):𝐴1-1𝐶) → 𝐴𝐶)
1513, 14mp3anl1 1455 . . . . . . 7 (((𝐴 ∈ V ∧ 𝐶 ∈ V) ∧ (𝑓𝐴):𝐴1-1𝐶) → 𝐴𝐶)
1615ancoms 459 . . . . . 6 (((𝑓𝐴):𝐴1-1𝐶 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐶)
1711, 16sylan 580 . . . . 5 (((𝑓:𝐵1-1𝐶𝐴𝐵) ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐶)
1817expl 458 . . . 4 (𝑓:𝐵1-1𝐶 → ((𝐴𝐵 ∧ (𝐴 ∈ V ∧ 𝐶 ∈ V)) → 𝐴𝐶))
1918exlimiv 1933 . . 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 396  wex 1781  wcel 2106  Vcvv 3445  wss 3910   class class class wbr 5105  cres 5635  1-1wf1 6493  cdom 8881
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 5256  ax-nul 5263  ax-pr 5384
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 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-dom 8885
This theorem is referenced by:  ssct  8995  1sdom2dom  9191
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