Theorem onsetreclem3 48051

Description: Lemma for onsetrec 48052. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)

Hypothesis

Ref	Expression
onsetreclem3.1	⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})

Assertion

Ref	Expression
onsetreclem3	⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))

Distinct variable group:   𝑥,𝑎

Allowed substitution hints:   𝐹(𝑥,𝑎)

Proof of Theorem onsetreclem3

Step	Hyp	Ref	Expression
1		eloni 6373	. . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎)
2		orduniorsuc 7825	. . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
3	1, 2	syl 17	. . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
4		vex 3473	. . . 4 ⊢ 𝑎 ∈ V
5	4	elpr 4647	. . 3 ⊢ (𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎} ↔ (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
6	3, 5	sylibr 233	. 2 ⊢ (𝑎 ∈ On → 𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎})
7		onsetreclem3.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
8	7	onsetreclem1 48049	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
9	6, 8	eleqtrrdi 2839	1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1534   ∈ wcel 2099  Vcvv 3469  {cpr 4626  ∪ cuni 4903   ↦ cmpt 5225  Ord word 6362  Oncon0 6363  suc csuc 6365  ‘cfv 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fv 6550

This theorem is referenced by:  onsetrec  48052