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Theorem onsetreclem1 42776
Description: Lemma for onsetrec 42779. (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 vex 3234 . 2 𝑎 ∈ V
2 unieq 4476 . . . 4 (𝑥 = 𝑎 𝑥 = 𝑎)
3 suceq 5828 . . . . 5 ( 𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
42, 3syl 17 . . . 4 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
52, 4preq12d 4308 . . 3 (𝑥 = 𝑎 → { 𝑥, suc 𝑥} = { 𝑎, suc 𝑎})
6 onsetreclem1.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
7 prex 4939 . . 3 { 𝑎, suc 𝑎} ∈ V
85, 6, 7fvmpt 6321 . 2 (𝑎 ∈ V → (𝐹𝑎) = { 𝑎, suc 𝑎})
91, 8ax-mp 5 1 (𝐹𝑎) = { 𝑎, suc 𝑎}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  Vcvv 3231  {cpr 4212   cuni 4468  cmpt 4762  suc csuc 5763  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  onsetreclem2  42777  onsetreclem3  42778
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