Theorem onsetreclem1 46296

Description: Lemma for onsetrec 46299. (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	⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}

Distinct variable group:   𝑥,𝑎

Allowed substitution hints:   𝐹(𝑥,𝑎)

Proof of Theorem onsetreclem1

Step	Hyp	Ref	Expression
1		unieq 4847	. . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎)
2		suceq 6316	. . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
3	1, 2	syl 17	. . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
4	1, 3	preq12d 4674	. . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎})
5		onsetreclem1.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
6		prex 5350	. . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V
7	4, 5, 6	fvmpt 6857	. 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎})
8	7	elv 3428	1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}

Syntax hints:   = wceq 1539  Vcvv 3422  {cpr 4560  ∪ cuni 4836   ↦ cmpt 5153  suc csuc 6253  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-suc 6257  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  onsetreclem2  46297  onsetreclem3  46298