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Mirrors > Home > MPE Home > Th. List > onovuni | Structured version Visualization version GIF version |
Description: A variant of onfununi 7978 for operations. (Contributed by Eric Schmidt, 26-May-2009.) (Revised by Mario Carneiro, 11-Sep-2015.) |
Ref | Expression |
---|---|
onovuni.1 | ⊢ (Lim 𝑦 → (𝐴𝐹𝑦) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥)) |
onovuni.2 | ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦)) |
Ref | Expression |
---|---|
onovuni | ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onovuni.1 | . . . 4 ⊢ (Lim 𝑦 → (𝐴𝐹𝑦) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥)) | |
2 | oveq2 7164 | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝐴𝐹𝑧) = (𝐴𝐹𝑦)) | |
3 | eqid 2821 | . . . . . 6 ⊢ (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) = (𝑧 ∈ V ↦ (𝐴𝐹𝑧)) | |
4 | ovex 7189 | . . . . . 6 ⊢ (𝐴𝐹𝑦) ∈ V | |
5 | 2, 3, 4 | fvmpt 6768 | . . . . 5 ⊢ (𝑦 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦)) |
6 | 5 | elv 3499 | . . . 4 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = (𝐴𝐹𝑦) |
7 | oveq2 7164 | . . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝐴𝐹𝑧) = (𝐴𝐹𝑥)) | |
8 | ovex 7189 | . . . . . . . 8 ⊢ (𝐴𝐹𝑥) ∈ V | |
9 | 7, 3, 8 | fvmpt 6768 | . . . . . . 7 ⊢ (𝑥 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)) |
10 | 9 | elv 3499 | . . . . . 6 ⊢ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥) |
11 | 10 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)) |
12 | 11 | iuneq2i 4940 | . . . 4 ⊢ ∪ 𝑥 ∈ 𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑦 (𝐴𝐹𝑥) |
13 | 1, 6, 12 | 3eqtr4g 2881 | . . 3 ⊢ (Lim 𝑦 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦) = ∪ 𝑥 ∈ 𝑦 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥)) |
14 | onovuni.2 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → (𝐴𝐹𝑥) ⊆ (𝐴𝐹𝑦)) | |
15 | 14, 10, 6 | 3sstr4g 4012 | . . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥 ⊆ 𝑦) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) ⊆ ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑦)) |
16 | 13, 15 | onfununi 7978 | . 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥)) |
17 | uniexg 7466 | . . . 4 ⊢ (𝑆 ∈ 𝑇 → ∪ 𝑆 ∈ V) | |
18 | oveq2 7164 | . . . . 5 ⊢ (𝑧 = ∪ 𝑆 → (𝐴𝐹𝑧) = (𝐴𝐹∪ 𝑆)) | |
19 | ovex 7189 | . . . . 5 ⊢ (𝐴𝐹∪ 𝑆) ∈ V | |
20 | 18, 3, 19 | fvmpt 6768 | . . . 4 ⊢ (∪ 𝑆 ∈ V → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = (𝐴𝐹∪ 𝑆)) |
21 | 17, 20 | syl 17 | . . 3 ⊢ (𝑆 ∈ 𝑇 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = (𝐴𝐹∪ 𝑆)) |
22 | 21 | 3ad2ant1 1129 | . 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘∪ 𝑆) = (𝐴𝐹∪ 𝑆)) |
23 | 10 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ 𝑆 → ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = (𝐴𝐹𝑥)) |
24 | 23 | iuneq2i 4940 | . . 3 ⊢ ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥) |
25 | 24 | a1i 11 | . 2 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∪ 𝑥 ∈ 𝑆 ((𝑧 ∈ V ↦ (𝐴𝐹𝑧))‘𝑥) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥)) |
26 | 16, 22, 25 | 3eqtr3d 2864 | 1 ⊢ ((𝑆 ∈ 𝑇 ∧ 𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐴𝐹∪ 𝑆) = ∪ 𝑥 ∈ 𝑆 (𝐴𝐹𝑥)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 ∪ cuni 4838 ∪ ciun 4919 ↦ cmpt 5146 Oncon0 6191 Lim wlim 6192 ‘cfv 6355 (class class class)co 7156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-ord 6194 df-on 6195 df-lim 6196 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 |
This theorem is referenced by: onoviun 7980 |
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