tz9.12lem2 - Metamath Proof Explorer

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Theorem tz9.12lem2 9809

Description: Lemma for tz9.12 9811. (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 9808	. . 3 ⊢ (𝐹 “ 𝐴) ⊆ On
4	2	funmpt2 6586	. . . . 5 ⊢ Fun 𝐹
5	1	funimaex 6635	. . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ 𝐴) ∈ V)
6	4, 5	ax-mp 5	. . . 4 ⊢ (𝐹 “ 𝐴) ∈ V
7	6	ssonunii 7780	. . 3 ⊢ ((𝐹 “ 𝐴) ⊆ On → ∪ (𝐹 “ 𝐴) ∈ On)
8	3, 7	ax-mp 5	. 2 ⊢ ∪ (𝐹 “ 𝐴) ∈ On
9	8	onsuci 7839	1 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On

Colors of variables: wff setvar class

Syntax hints: = wceq 1533 ∈ wcel 2098 {crab 3419 Vcvv 3463 ⊆ wss 3940 ∪ cuni 4903 ∩ cint 4944 ↦ cmpt 5226 “ cima 5675 Oncon0 6364 suc csuc 6366 Fun wfun 6536 ‘cfv 6542 𝑅₁cr1 9783

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 5280 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7737

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 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ord 6367 df-on 6368 df-suc 6370 df-fun 6544

This theorem is referenced by: tz9.12lem3 9810 tz9.12 9811

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