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Mirrors > Home > MPE Home > Th. List > tz9.12lem2 | Structured version Visualization version GIF version |
Description: Lemma for tz9.12 9828. (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 9825 | . . 3 ⊢ (𝐹 “ 𝐴) ⊆ On |
4 | 2 | funmpt2 6607 | . . . . 5 ⊢ Fun 𝐹 |
5 | 1 | funimaex 6656 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ 𝐴) ∈ V) |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝐹 “ 𝐴) ∈ V |
7 | 6 | ssonunii 7800 | . . 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 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ⊆ wss 3963 ∪ cuni 4912 ∩ cint 4951 ↦ cmpt 5231 “ cima 5692 Oncon0 6386 suc csuc 6388 Fun wfun 6557 ‘cfv 6563 𝑅1cr1 9800 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-fun 6565 |
This theorem is referenced by: tz9.12lem3 9827 tz9.12 9828 |
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