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Theorem onsetreclem1 48244
Description: Lemma for onsetrec 48247. (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 4915 . . . 4 (𝑥 = 𝑎 𝑥 = 𝑎)
2 suceq 6431 . . . . 5 ( 𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
31, 2syl 17 . . . 4 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
41, 3preq12d 4742 . . 3 (𝑥 = 𝑎 → { 𝑥, suc 𝑥} = { 𝑎, suc 𝑎})
5 onsetreclem1.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
6 prex 5429 . . 3 { 𝑎, suc 𝑎} ∈ V
74, 5, 6fvmpt 6998 . 2 (𝑎 ∈ V → (𝐹𝑎) = { 𝑎, suc 𝑎})
87elv 3469 1 (𝐹𝑎) = { 𝑎, suc 𝑎}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  Vcvv 3463  {cpr 4627   cuni 4904  cmpt 5227  suc csuc 6367  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-suc 6371  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  onsetreclem2  48245  onsetreclem3  48246
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