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Theorem harword 9601
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 9046 . . . . 5 ((𝑦𝑋𝑋𝑌) → 𝑦𝑌)
21expcom 413 . . . 4 (𝑋𝑌 → (𝑦𝑋𝑦𝑌))
32adantr 480 . . 3 ((𝑋𝑌𝑦 ∈ On) → (𝑦𝑋𝑦𝑌))
43ss2rabdv 4086 . 2 (𝑋𝑌 → {𝑦 ∈ On ∣ 𝑦𝑋} ⊆ {𝑦 ∈ On ∣ 𝑦𝑌})
5 reldom 8990 . . . 4 Rel ≼
65brrelex1i 5745 . . 3 (𝑋𝑌𝑋 ∈ V)
7 harval 9598 . . 3 (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
86, 7syl 17 . 2 (𝑋𝑌 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
95brrelex2i 5746 . . 3 (𝑋𝑌𝑌 ∈ V)
10 harval 9598 . . 3 (𝑌 ∈ V → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦𝑌})
119, 10syl 17 . 2 (𝑋𝑌 → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦𝑌})
124, 8, 113sstr4d 4043 1 (𝑋𝑌 → (har‘𝑋) ⊆ (har‘𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {crab 3433  Vcvv 3478  wss 3963   class class class wbr 5148  Oncon0 6386  cfv 6563  cdom 8982  harchar 9594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-en 8985  df-dom 8986  df-oi 9548  df-har 9595
This theorem is referenced by:  hsmexlem3  10466
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