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Theorem harval 8350
 Description: Function value of the Hartogs function. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
harval (𝑋𝑉 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
Distinct variable group:   𝑦,𝑋
Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem harval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝑋𝑉𝑋 ∈ V)
2 breq2 4587 . . . 4 (𝑥 = 𝑋 → (𝑦𝑥𝑦𝑋))
32rabbidv 3164 . . 3 (𝑥 = 𝑋 → {𝑦 ∈ On ∣ 𝑦𝑥} = {𝑦 ∈ On ∣ 𝑦𝑋})
4 df-har 8346 . . 3 har = (𝑥 ∈ V ↦ {𝑦 ∈ On ∣ 𝑦𝑥})
5 hartogs 8332 . . 3 (𝑥 ∈ V → {𝑦 ∈ On ∣ 𝑦𝑥} ∈ On)
63, 4, 5fvmpt3 6195 . 2 (𝑋 ∈ V → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
71, 6syl 17 1 (𝑋𝑉 → (har‘𝑋) = {𝑦 ∈ On ∣ 𝑦𝑋})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583  Oncon0 5640  ‘cfv 5804   ≼ cdom 7839  harchar 8344 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-wrecs 7294  df-recs 7355  df-en 7842  df-dom 7843  df-oi 8298  df-har 8346 This theorem is referenced by:  elharval  8351  harword  8353  harwdom  8378  harval2  8706
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