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Theorem harword 9024
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 8558 . . . . 5 ((𝑦𝑋𝑋𝑌) → 𝑦𝑌)
21expcom 417 . . . 4 (𝑋𝑌 → (𝑦𝑋𝑦𝑌))
32adantr 484 . . 3 ((𝑋𝑌𝑦 ∈ On) → (𝑦𝑋𝑦𝑌))
43ss2rabdv 4038 . 2 (𝑋𝑌 → {𝑦 ∈ On ∣ 𝑦𝑋} ⊆ {𝑦 ∈ On ∣ 𝑦𝑌})
5 reldom 8511 . . . 4 Rel ≼
65brrelex1i 5595 . . 3 (𝑋𝑌𝑋 ∈ V)
7 harval 9021 . . 3 (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
86, 7syl 17 . 2 (𝑋𝑌 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
95brrelex2i 5596 . . 3 (𝑋𝑌𝑌 ∈ V)
10 harval 9021 . . 3 (𝑌 ∈ V → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦𝑌})
119, 10syl 17 . 2 (𝑋𝑌 → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦𝑌})
124, 8, 113sstr4d 4000 1 (𝑋𝑌 → (har‘𝑋) ⊆ (har‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2115  {crab 3137  Vcvv 3480  wss 3919   class class class wbr 5052  Oncon0 6178  cfv 6343  cdom 8503  harchar 9017
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-se 5502  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-isom 6352  df-riota 7107  df-wrecs 7943  df-recs 8004  df-en 8506  df-dom 8507  df-oi 8971  df-har 9018
This theorem is referenced by:  hsmexlem3  9848
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