Theorem onsetreclem1 49536

Description: Lemma for onsetrec 49539. (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 4899	. . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎)
2		suceq 6424	. . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
3	1, 2	syl 17	. . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
4	1, 3	preq12d 4722	. . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎})
5		onsetreclem1.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
6		prex 5412	. . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V
7	4, 5, 6	fvmpt 6991	. 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎})
8	7	elv 3469	1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}

Syntax hints:   = wceq 1540  Vcvv 3464  {cpr 4608  ∪ cuni 4888   ↦ cmpt 5206  suc csuc 6359  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-suc 6363  df-iota 6489  df-fun 6538  df-fv 6544

This theorem is referenced by:  onsetreclem2  49537  onsetreclem3  49538