Theorem onsetreclem3 44816

Description: Lemma for onsetrec 44817. (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 6203	. . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎)
2		orduniorsuc 7547	. . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
3	1, 2	syl 17	. . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
4		vex 3499	. . . 4 ⊢ 𝑎 ∈ V
5	4	elpr 4592	. . 3 ⊢ (𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎} ↔ (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
6	3, 5	sylibr 236	. 2 ⊢ (𝑎 ∈ On → 𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎})
7		onsetreclem3.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
8	7	onsetreclem1 44814	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
9	6, 8	eleqtrrdi 2926	1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))

Syntax hints:   → wi 4   ∨ wo 843   = wceq 1537   ∈ wcel 2114  Vcvv 3496  {cpr 4571  ∪ cuni 4840   ↦ cmpt 5148  Ord word 6192  Oncon0 6193  suc csuc 6195  ‘cfv 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-ord 6196  df-on 6197  df-suc 6199  df-iota 6316  df-fun 6359  df-fv 6365

This theorem is referenced by:  onsetrec  44817