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Theorem harword 8712
Description: Weak ordering property of the Hartogs function. (Contributed by Stefan O'Rear, 14-Feb-2015.)
Assertion
Ref Expression
harword (𝑋𝑌 → (har‘𝑋) ⊆ (har‘𝑌))

Proof of Theorem harword
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 domtr 8248 . . . . 5 ((𝑦𝑋𝑋𝑌) → 𝑦𝑌)
21expcom 403 . . . 4 (𝑋𝑌 → (𝑦𝑋𝑦𝑌))
32adantr 473 . . 3 ((𝑋𝑌𝑦 ∈ On) → (𝑦𝑋𝑦𝑌))
43ss2rabdv 3879 . 2 (𝑋𝑌 → {𝑦 ∈ On ∣ 𝑦𝑋} ⊆ {𝑦 ∈ On ∣ 𝑦𝑌})
5 reldom 8201 . . . 4 Rel ≼
65brrelex1i 5363 . . 3 (𝑋𝑌𝑋 ∈ V)
7 harval 8709 . . 3 (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
86, 7syl 17 . 2 (𝑋𝑌 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
95brrelex2i 5364 . . 3 (𝑋𝑌𝑌 ∈ V)
10 harval 8709 . . 3 (𝑌 ∈ V → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦𝑌})
119, 10syl 17 . 2 (𝑋𝑌 → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦𝑌})
124, 8, 113sstr4d 3844 1 (𝑋𝑌 → (har‘𝑋) ⊆ (har‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  {crab 3093  Vcvv 3385  wss 3769   class class class wbr 4843  Oncon0 5941  cfv 6101  cdom 8193  harchar 8703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-tp 4373  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-tr 4946  df-id 5220  df-eprel 5225  df-po 5233  df-so 5234  df-fr 5271  df-se 5272  df-we 5273  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-pred 5898  df-ord 5944  df-on 5945  df-lim 5946  df-suc 5947  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-isom 6110  df-riota 6839  df-wrecs 7645  df-recs 7707  df-en 8196  df-dom 8197  df-oi 8657  df-har 8705
This theorem is referenced by:  hsmexlem3  9538
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