Theorem onsetreclem2 50512

Description: Lemma for onsetrec 50514. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)

Hypothesis

Ref	Expression
onsetreclem2.1	⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})

Assertion

Ref	Expression
onsetreclem2	⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)

Distinct variable group:   𝑥,𝑎

Allowed substitution hints:   𝐹(𝑥, 𝑎)

Proof of Theorem onsetreclem2

Step	Hyp	Ref	Expression
1		onsetreclem2.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
2	1	onsetreclem1 50511	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
3		vex 3458	. . . 4 ⊢ 𝑎 ∈ V
4	3	ssonunii 7778	. . 3 ⊢ (𝑎 ⊆ On → ∪ 𝑎 ∈ On)
5		onsuc 7807	. . 3 ⊢ (∪ 𝑎 ∈ On → suc ∪ 𝑎 ∈ On)
6		prssi 4786	. . 3 ⊢ ((∪ 𝑎 ∈ On ∧ suc ∪ 𝑎 ∈ On) → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
7	4, 5, 6	syl2anc2 596	. 2 ⊢ (𝑎 ⊆ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
8	2, 7	eqsstrid 3974	1 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  Vcvv 3454   ⊆ wss 3904  {cpr 4590  ∪ cuni 4871   ↦ cmpt 5191  Oncon0 6360  suc csuc 6362  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fv 6544

This theorem is used by:  onsetrec  50514