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Theorem onsetreclem2 48937
Description: Lemma for onsetrec 48939. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
Hypothesis
Ref Expression
onsetreclem2.1 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
Assertion
Ref Expression
onsetreclem2 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
Distinct variable group:   𝑥,𝑎
Allowed substitution hints:   𝐹(𝑥,𝑎)

Proof of Theorem onsetreclem2
StepHypRef Expression
1 onsetreclem2.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
21onsetreclem1 48936 . 2 (𝐹𝑎) = { 𝑎, suc 𝑎}
3 vex 3482 . . . 4 𝑎 ∈ V
43ssonunii 7800 . . 3 (𝑎 ⊆ On → 𝑎 ∈ On)
5 onsuc 7831 . . 3 ( 𝑎 ∈ On → suc 𝑎 ∈ On)
6 prssi 4826 . . 3 (( 𝑎 ∈ On ∧ suc 𝑎 ∈ On) → { 𝑎, suc 𝑎} ⊆ On)
74, 5, 6syl2anc2 585 . 2 (𝑎 ⊆ On → { 𝑎, suc 𝑎} ⊆ On)
82, 7eqsstrid 4044 1 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  wss 3963  {cpr 4633   cuni 4912  cmpt 5231  Oncon0 6386  suc csuc 6388  cfv 6563
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-sep 5302  ax-nul 5312  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-rab 3434  df-v 3480  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-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-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571
This theorem is referenced by:  onsetrec  48939
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