Theorem onsetreclem1 48797

Description: Lemma for onsetrec 48800. (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 4942	. . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎)
2		suceq 6461	. . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
3	1, 2	syl 17	. . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
4	1, 3	preq12d 4766	. . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎})
5		onsetreclem1.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
6		prex 5452	. . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V
7	4, 5, 6	fvmpt 7029	. 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎})
8	7	elv 3493	1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}

Syntax hints:   = wceq 1537  Vcvv 3488  {cpr 4650  ∪ cuni 4931   ↦ cmpt 5249  suc csuc 6397  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-suc 6401  df-iota 6525  df-fun 6575  df-fv 6581

This theorem is referenced by:  onsetreclem2  48798  onsetreclem3  48799