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Theorem harword 8467
 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 8006 . . . . 5 ((𝑦𝑋𝑋𝑌) → 𝑦𝑌)
21expcom 451 . . . 4 (𝑋𝑌 → (𝑦𝑋𝑦𝑌))
32adantr 481 . . 3 ((𝑋𝑌𝑦 ∈ On) → (𝑦𝑋𝑦𝑌))
43ss2rabdv 3681 . 2 (𝑋𝑌 → {𝑦 ∈ On ∣ 𝑦𝑋} ⊆ {𝑦 ∈ On ∣ 𝑦𝑌})
5 reldom 7958 . . . 4 Rel ≼
65brrelexi 5156 . . 3 (𝑋𝑌𝑋 ∈ V)
7 harval 8464 . . 3 (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
86, 7syl 17 . 2 (𝑋𝑌 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
95brrelex2i 5157 . . 3 (𝑋𝑌𝑌 ∈ V)
10 harval 8464 . . 3 (𝑌 ∈ V → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦𝑌})
119, 10syl 17 . 2 (𝑋𝑌 → (har‘𝑌) = {𝑦 ∈ On ∣ 𝑦𝑌})
124, 8, 113sstr4d 3646 1 (𝑋𝑌 → (har‘𝑋) ⊆ (har‘𝑌))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1482   ∈ wcel 1989  {crab 2915  Vcvv 3198   ⊆ wss 3572   class class class wbr 4651  Oncon0 5721  ‘cfv 5886   ≼ cdom 7950  harchar 8458 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rmo 2919  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-se 5072  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-isom 5895  df-riota 6608  df-wrecs 7404  df-recs 7465  df-en 7953  df-dom 7954  df-oi 8412  df-har 8460 This theorem is referenced by:  hsmexlem3  9247
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