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Theorem onsetreclem2 49280
Description: Lemma for onsetrec 49282. (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 49279 . 2 (𝐹𝑎) = { 𝑎, suc 𝑎}
3 vex 3483 . . . 4 𝑎 ∈ V
43ssonunii 7802 . . 3 (𝑎 ⊆ On → 𝑎 ∈ On)
5 onsuc 7832 . . 3 ( 𝑎 ∈ On → suc 𝑎 ∈ On)
6 prssi 4820 . . 3 (( 𝑎 ∈ On ∧ suc 𝑎 ∈ On) → { 𝑎, suc 𝑎} ⊆ On)
74, 5, 6syl2anc2 585 . 2 (𝑎 ⊆ On → { 𝑎, suc 𝑎} ⊆ On)
82, 7eqsstrid 4021 1 (𝑎 ⊆ On → (𝐹𝑎) ⊆ On)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2107  Vcvv 3479  wss 3950  {cpr 4627   cuni 4906  cmpt 5224  Oncon0 6383  suc csuc 6385  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-ord 6386  df-on 6387  df-suc 6389  df-iota 6513  df-fun 6562  df-fv 6568
This theorem is referenced by:  onsetrec  49282
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