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Theorem onsetreclem1 47703
Description: Lemma for onsetrec 47706. (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 4918 . . . 4 (𝑥 = 𝑎 𝑥 = 𝑎)
2 suceq 6427 . . . . 5 ( 𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
31, 2syl 17 . . . 4 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
41, 3preq12d 4744 . . 3 (𝑥 = 𝑎 → { 𝑥, suc 𝑥} = { 𝑎, suc 𝑎})
5 onsetreclem1.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
6 prex 5431 . . 3 { 𝑎, suc 𝑎} ∈ V
74, 5, 6fvmpt 6995 . 2 (𝑎 ∈ V → (𝐹𝑎) = { 𝑎, suc 𝑎})
87elv 3480 1 (𝐹𝑎) = { 𝑎, suc 𝑎}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541  Vcvv 3474  {cpr 4629   cuni 4907  cmpt 5230  suc csuc 6363  cfv 6540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-suc 6367  df-iota 6492  df-fun 6542  df-fv 6548
This theorem is referenced by:  onsetreclem2  47704  onsetreclem3  47705
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