Theorem onsetreclem3 47705

Description: Lemma for onsetrec 47706. (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 6371	. . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎)
2		orduniorsuc 7814	. . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
3	1, 2	syl 17	. . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
4		vex 3478	. . . 4 ⊢ 𝑎 ∈ V
5	4	elpr 4650	. . 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 47703	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
9	6, 8	eleqtrrdi 2844	1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))

Syntax hints:   → wi 4   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3474  {cpr 4629  ∪ cuni 4907   ↦ cmpt 5230  Ord word 6360  Oncon0 6361  suc csuc 6363  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6492  df-fun 6542  df-fv 6548

This theorem is referenced by:  onsetrec  47706