tz9.12lem2 - Metamath Proof Explorer

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Theorem tz9.12lem2 9813

Description: Lemma for tz9.12 9815. (Contributed by NM, 22-Sep-2003.)

**Hypotheses**
Ref	Expression
tz9.12lem.1	⊢ 𝐴 ∈ V
tz9.12lem.2	⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅₁‘𝑣)})

**Assertion**
Ref	Expression
tz9.12lem2	⊢ suc ∪ (𝐹 “ 𝐴) ∈ On

Distinct variable group: 𝑧,𝑣,𝐴 Allowed substitution hints: 𝐹(𝑧,𝑣)

**Proof of Theorem tz9.12lem2**
Step	Hyp	Ref	Expression
1		tz9.12lem.1	. . . 4 ⊢ 𝐴 ∈ V
2		tz9.12lem.2	. . . 4 ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅₁‘𝑣)})
3	1, 2	tz9.12lem1 9812	. . 3 ⊢ (𝐹 “ 𝐴) ⊆ On
4	2	funmpt2 6593	. . . . 5 ⊢ Fun 𝐹
5	1	funimaex 6642	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ 𝐴) ∈ V)
6	4, 5	ax-mp 5	. . . 4 ⊢ (𝐹 “ 𝐴) ∈ V
7	6	ssonunii 7784	. . 3 ⊢ ((𝐹 “ 𝐴) ⊆ On → ∪ (𝐹 “ 𝐴) ∈ On)
8	3, 7	ax-mp 5	. 2 ⊢ ∪ (𝐹 “ 𝐴) ∈ On
9	8	onsuci 7843	1 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 {crab 3418 Vcvv 3461 ⊆ wss 3944 ∪ cuni 4909 ∩ cint 4950 ↦ cmpt 5232 “ cima 5681 Oncon0 6371 suc csuc 6373 Fun wfun 6543 ‘cfv 6549 𝑅₁cr1 9787

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5429 ax-un 7741

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-suc 6377 df-fun 6551

This theorem is referenced by: tz9.12lem3 9814 tz9.12 9815

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