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| Mirrors > Home > MPE Home > Th. List > tz9.12lem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for tz9.12 9830. (Contributed by NM, 22-Sep-2003.) |
| Ref | Expression |
|---|---|
| tz9.12lem.1 | ⊢ 𝐴 ∈ V |
| tz9.12lem.2 | ⊢ 𝐹 = (𝑧 ∈ V ↦ ∩ {𝑣 ∈ On ∣ 𝑧 ∈ (𝑅1‘𝑣)}) |
| Ref | Expression |
|---|---|
| tz9.12lem2 | ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On |
| 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 9827 | . . 3 ⊢ (𝐹 “ 𝐴) ⊆ On |
| 4 | 2 | funmpt2 6605 | . . . . 5 ⊢ Fun 𝐹 |
| 5 | 1 | funimaex 6655 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ 𝐴) ∈ V) |
| 6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝐹 “ 𝐴) ∈ V |
| 7 | 6 | ssonunii 7801 | . . 3 ⊢ ((𝐹 “ 𝐴) ⊆ On → ∪ (𝐹 “ 𝐴) ∈ On) |
| 8 | 3, 7 | ax-mp 5 | . 2 ⊢ ∪ (𝐹 “ 𝐴) ∈ On |
| 9 | 8 | onsuci 7859 | 1 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ⊆ wss 3951 ∪ cuni 4907 ∩ cint 4946 ↦ cmpt 5225 “ cima 5688 Oncon0 6384 suc csuc 6386 Fun wfun 6555 ‘cfv 6561 𝑅1cr1 9802 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-suc 6390 df-fun 6563 |
| This theorem is referenced by: tz9.12lem3 9829 tz9.12 9830 |
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