Theorem onsetreclem2 50634

Description: Lemma for onsetrec 50636. (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 50633	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
3		vex 3457	. . . 4 ⊢ 𝑎 ∈ V
4	3	ssonunii 7783	. . 3 ⊢ (𝑎 ⊆ On → ∪ 𝑎 ∈ On)
5		onsuc 7812	. . 3 ⊢ (∪ 𝑎 ∈ On → suc ∪ 𝑎 ∈ On)
6		prssi 4785	. . 3 ⊢ ((∪ 𝑎 ∈ On ∧ suc ∪ 𝑎 ∈ On) → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
7	4, 5, 6	syl2anc2 597	. 2 ⊢ (𝑎 ⊆ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
8	2, 7	eqsstrid 3972	1 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3453   ⊆ wss 3902  {cpr 4589  ∪ cuni 4870   ↦ cmpt 5190  Oncon0 6361  suc csuc 6363  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402  ax-un 7739

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fv 6545

This theorem is used by:  onsetrec  50636