Theorem onsetreclem1 50326

Description: Lemma for onsetrec 50329. (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 4876	. . . 4 ⊢ (𝑥 = 𝑎 → ∪ 𝑥 = ∪ 𝑎)
2		suceq 6414	. . . . 5 ⊢ (∪ 𝑥 = ∪ 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
3	1, 2	syl 17	. . . 4 ⊢ (𝑥 = 𝑎 → suc ∪ 𝑥 = suc ∪ 𝑎)
4	1, 3	preq12d 4700	. . 3 ⊢ (𝑥 = 𝑎 → {∪ 𝑥, suc ∪ 𝑥} = {∪ 𝑎, suc ∪ 𝑎})
5		onsetreclem1.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
6		prex 5395	. . 3 ⊢ {∪ 𝑎, suc ∪ 𝑎} ∈ V
7	4, 5, 6	fvmpt 6975	. 2 ⊢ (𝑎 ∈ V → (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎})
8	7	elv 3459	1 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}

Syntax hints:   = wceq 1560  Vcvv 3454  {cpr 4584  ∪ cuni 4865   ↦ cmpt 5181  suc csuc 6348  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-suc 6352  df-iota 6477  df-fun 6523  df-fv 6529

This theorem is referenced by:  onsetreclem2  50327  onsetreclem3  50328