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Mirrors > Home > MPE Home > Th. List > tz9.12lem2 | Structured version Visualization version GIF version |
Description: Lemma for tz9.12 8536. (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 8533 | . . 3 ⊢ (𝐹 “ 𝐴) ⊆ On |
4 | 2 | funmpt2 5841 | . . . . 5 ⊢ Fun 𝐹 |
5 | 1 | funimaex 5890 | . . . . 5 ⊢ (Fun 𝐹 → (𝐹 “ 𝐴) ∈ V) |
6 | 4, 5 | ax-mp 5 | . . . 4 ⊢ (𝐹 “ 𝐴) ∈ V |
7 | 6 | ssonunii 6879 | . . 3 ⊢ ((𝐹 “ 𝐴) ⊆ On → ∪ (𝐹 “ 𝐴) ∈ On) |
8 | 3, 7 | ax-mp 5 | . 2 ⊢ ∪ (𝐹 “ 𝐴) ∈ On |
9 | 8 | onsuci 6930 | 1 ⊢ suc ∪ (𝐹 “ 𝐴) ∈ On |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∪ cuni 4372 ∩ cint 4410 ↦ cmpt 4643 “ cima 5041 Oncon0 5640 suc csuc 5642 Fun wfun 5798 ‘cfv 5804 𝑅1cr1 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ord 5643 df-on 5644 df-suc 5646 df-fun 5806 |
This theorem is referenced by: tz9.12lem3 8535 tz9.12 8536 |
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