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Theorem onsetreclem1 46081
Description: Lemma for onsetrec 46084. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)
Hypothesis
Ref Expression
onsetreclem1.1 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
Assertion
Ref Expression
onsetreclem1 (𝐹𝑎) = { 𝑎, suc 𝑎}
Distinct variable group:   𝑥,𝑎
Allowed substitution hints:   𝐹(𝑥,𝑎)

Proof of Theorem onsetreclem1
StepHypRef Expression
1 unieq 4830 . . . 4 (𝑥 = 𝑎 𝑥 = 𝑎)
2 suceq 6278 . . . . 5 ( 𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
31, 2syl 17 . . . 4 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
41, 3preq12d 4657 . . 3 (𝑥 = 𝑎 → { 𝑥, suc 𝑥} = { 𝑎, suc 𝑎})
5 onsetreclem1.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
6 prex 5325 . . 3 { 𝑎, suc 𝑎} ∈ V
74, 5, 6fvmpt 6818 . 2 (𝑎 ∈ V → (𝐹𝑎) = { 𝑎, suc 𝑎})
87elv 3414 1 (𝐹𝑎) = { 𝑎, suc 𝑎}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1543  Vcvv 3408  {cpr 4543   cuni 4819  cmpt 5135  suc csuc 6215  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-suc 6219  df-iota 6338  df-fun 6382  df-fv 6388
This theorem is referenced by:  onsetreclem2  46082  onsetreclem3  46083
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