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Theorem onsetreclem1 42969
Description: Lemma for onsetrec 42972. (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 3352 . 2 𝑎 ∈ V
2 unieq 4580 . . . 4 (𝑥 = 𝑎 𝑥 = 𝑎)
3 suceq 5933 . . . . 5 ( 𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
42, 3syl 17 . . . 4 (𝑥 = 𝑎 → suc 𝑥 = suc 𝑎)
52, 4preq12d 4410 . . 3 (𝑥 = 𝑎 → { 𝑥, suc 𝑥} = { 𝑎, suc 𝑎})
6 onsetreclem1.1 . . 3 𝐹 = (𝑥 ∈ V ↦ { 𝑥, suc 𝑥})
7 prex 5037 . . 3 { 𝑎, suc 𝑎} ∈ V
85, 6, 7fvmpt 6424 . 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 1630  wcel 2144  Vcvv 3349  {cpr 4316   cuni 4572  cmpt 4861  suc csuc 5868  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-suc 5872  df-iota 5994  df-fun 6033  df-fv 6039
This theorem is referenced by:  onsetreclem2  42970  onsetreclem3  42971
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