Theorem onsetreclem2 46297

Description: Lemma for onsetrec 46299. (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 46296	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
3		vex 3426	. . . 4 ⊢ 𝑎 ∈ V
4	3	ssonunii 7608	. . 3 ⊢ (𝑎 ⊆ On → ∪ 𝑎 ∈ On)
5		suceloni 7635	. . 3 ⊢ (∪ 𝑎 ∈ On → suc ∪ 𝑎 ∈ On)
6		prssi 4751	. . 3 ⊢ ((∪ 𝑎 ∈ On ∧ suc ∪ 𝑎 ∈ On) → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
7	4, 5, 6	syl2anc2 584	. 2 ⊢ (𝑎 ⊆ On → {∪ 𝑎, suc ∪ 𝑎} ⊆ On)
8	2, 7	eqsstrid 3965	1 ⊢ (𝑎 ⊆ On → (𝐹‘𝑎) ⊆ On)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ⊆ wss 3883  {cpr 4560  ∪ cuni 4836   ↦ cmpt 5153  Oncon0 6251  suc csuc 6253  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-ord 6254  df-on 6255  df-suc 6257  df-iota 6376  df-fun 6420  df-fv 6426

This theorem is referenced by:  onsetrec  46299