Theorem onsetreclem2 45699

Description: Lemma for onsetrec 45701. (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 45698	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
3		vex 3413	. . . 4 ⊢ 𝑎 ∈ V
4	3	ssonunii 7506	. . 3 ⊢ (𝑎 ⊆ On → ∪ 𝑎 ∈ On)
5		suceloni 7532	. . 3 ⊢ (∪ 𝑎 ∈ On → suc ∪ 𝑎 ∈ On)
6		prssi 4714	. . 3 ⊢ ((∪ 𝑎 ∈ On ∧ suc ∪ 𝑎 ∈ On) → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
7	4, 5, 6	syl2anc2 588	. 2 ⊢ (𝑎 ⊆ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
8	2, 7	eqsstrid 3942	1 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3860  {cpr 4527  ∪ cuni 4801   ↦ cmpt 5115  Oncon0 6173  suc csuc 6175  ‘cfv 6339

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301  ax-un 7464

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-ord 6176  df-on 6177  df-suc 6179  df-iota 6298  df-fun 6341  df-fv 6347

This theorem is referenced by:  onsetrec  45701