Theorem onsetreclem2 44209

Description: Lemma for onsetrec 44211. (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 44208	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
3		vex 3411	. . . 4 ⊢ 𝑎 ∈ V
4	3	ssonunii 7316	. . 3 ⊢ (𝑎 ⊆ On → ∪ 𝑎 ∈ On)
5		suceloni 7342	. . 3 ⊢ (∪ 𝑎 ∈ On → suc ∪ 𝑎 ∈ On)
6		prssi 4624	. . 3 ⊢ ((∪ 𝑎 ∈ On ∧ suc ∪ 𝑎 ∈ On) → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
7	4, 5, 6	syl2anc2 577	. 2 ⊢ (𝑎 ⊆ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
8	2, 7	syl5eqss 3898	1 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)

Syntax hints:   → wi 4   = wceq 1508   ∈ wcel 2051  Vcvv 3408   ⊆ wss 3822  {cpr 4437  ∪ cuni 4708   ↦ cmpt 5004  Oncon0 6026  suc csuc 6028  ‘cfv 6185

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-sep 5056  ax-nul 5063  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-ord 6029  df-on 6030  df-suc 6032  df-iota 6149  df-fun 6187  df-fv 6193

This theorem is referenced by:  onsetrec  44211