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Mirrors > Home > MPE Home > Th. List > tz9.12lem2 | Structured version Visualization version GIF version |
Description: Lemma for tz9.12 9213. (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 9210 | . . 3 ⊢ (𝐹 “ 𝐴) ⊆ On |
4 | 2 | funmpt2 6389 | . . . . 5 ⊢ Fun 𝐹 |
5 | 1 | funimaex 6436 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ 𝐴) ∈ V) |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝐹 “ 𝐴) ∈ V |
7 | 6 | ssonunii 7496 | . . 3 ⊢ ((𝐹 “ 𝐴) ⊆ On → ∪ (𝐹 “ 𝐴) ∈ On) |
8 | 3, 7 | ax-mp 5 | . 2 ⊢ ∪ (𝐹 “ 𝐴) ∈ On |
9 | 8 | onsuci 7547 | 1 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 {crab 3142 Vcvv 3495 ⊆ wss 3936 ∪ cuni 4832 ∩ cint 4869 ↦ cmpt 5139 “ cima 5553 Oncon0 6186 suc csuc 6188 Fun wfun 6344 ‘cfv 6350 𝑅1cr1 9185 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-suc 6192 df-fun 6352 |
This theorem is referenced by: tz9.12lem3 9212 tz9.12 9213 |
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