Theorem onsetreclem3 43558

Description: Lemma for onsetrec 43559. (Contributed by Emmett Weisz, 22-Jun-2021.) (New usage is discouraged.)

Hypothesis

Ref	Expression
onsetreclem3.1	⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})

Assertion

Ref	Expression
onsetreclem3	⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))

Distinct variable group:   𝑥,𝑎

Allowed substitution hints:   𝐹(𝑥,𝑎)

Proof of Theorem onsetreclem3

Step	Hyp	Ref	Expression
1		eloni 5986	. . . 4 ⊢ (𝑎 ∈ On → Ord 𝑎)
2		orduniorsuc 7308	. . . 4 ⊢ (Ord 𝑎 → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
3	1, 2	syl 17	. . 3 ⊢ (𝑎 ∈ On → (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
4		vex 3401	. . . 4 ⊢ 𝑎 ∈ V
5	4	elpr 4421	. . 3 ⊢ (𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎} ↔ (𝑎 = ∪ 𝑎 ∨ 𝑎 = suc ∪ 𝑎))
6	3, 5	sylibr 226	. 2 ⊢ (𝑎 ∈ On → 𝑎 ∈ {∪ 𝑎, suc ∪ 𝑎})
7		onsetreclem3.1	. . 3 ⊢ 𝐹 = (𝑥 ∈ V ↦ {∪ 𝑥, suc ∪ 𝑥})
8	7	onsetreclem1 43556	. 2 ⊢ (𝐹‘𝑎) = {∪ 𝑎, suc ∪ 𝑎}
9	6, 8	syl6eleqr 2870	1 ⊢ (𝑎 ∈ On → 𝑎 ∈ (𝐹‘𝑎))

Syntax hints:   → wi 4   ∨ wo 836   = wceq 1601   ∈ wcel 2107  Vcvv 3398  {cpr 4400  ∪ cuni 4671   ↦ cmpt 4965  Ord word 5975  Oncon0 5976  suc csuc 5978  ‘cfv 6135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138  ax-un 7226

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-ord 5979  df-on 5980  df-suc 5982  df-iota 6099  df-fun 6137  df-fv 6143

This theorem is referenced by:  onsetrec  43559