Theorem onsetreclem1 49279

Description: Lemma for onsetrec 49282. (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 4917	. . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎)
2		suceq 6449	. . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
3	1, 2	syl 17	. . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
4	1, 3	preq12d 4740	. . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎})
5		onsetreclem1.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
6		prex 5436	. . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V
7	4, 5, 6	fvmpt 7015	. 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎})
8	7	elv 3484	1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}

Syntax hints:   = wceq 1539  Vcvv 3479  {cpr 4627  ∪ cuni 4906   ↦ cmpt 5224  suc csuc 6385  ‘cfv 6560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-suc 6389  df-iota 6513  df-fun 6562  df-fv 6568

This theorem is referenced by:  onsetreclem2  49280  onsetreclem3  49281