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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > aomclem3 | Structured version Visualization version GIF version |
Description: Lemma for dfac11 41418. Successor case 3, our required well-ordering. (Contributed by Stefan O'Rear, 19-Jan-2015.) |
Ref | Expression |
---|---|
aomclem3.b | ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} |
aomclem3.c | ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) |
aomclem3.d | ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) |
aomclem3.e | ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} |
aomclem3.on | ⊢ (𝜑 → dom 𝑧 ∈ On) |
aomclem3.su | ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) |
aomclem3.we | ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) |
aomclem3.a | ⊢ (𝜑 → 𝐴 ∈ On) |
aomclem3.za | ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴) |
aomclem3.y | ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) |
Ref | Expression |
---|---|
aomclem3 | ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | aomclem3.d | . . 3 ⊢ 𝐷 = recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) | |
2 | rneq 5896 | . . . . . . 7 ⊢ (𝑎 = 𝑐 → ran 𝑎 = ran 𝑐) | |
3 | 2 | difeq2d 4087 | . . . . . 6 ⊢ (𝑎 = 𝑐 → ((𝑅1‘dom 𝑧) ∖ ran 𝑎) = ((𝑅1‘dom 𝑧) ∖ ran 𝑐)) |
4 | 3 | fveq2d 6851 | . . . . 5 ⊢ (𝑎 = 𝑐 → (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)) = (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))) |
5 | 4 | cbvmptv 5223 | . . . 4 ⊢ (𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))) |
6 | recseq 8325 | . . . 4 ⊢ ((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))) → recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))))) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ recs((𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)))) = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))) |
8 | 1, 7 | eqtri 2765 | . 2 ⊢ 𝐷 = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))) |
9 | fvexd 6862 | . 2 ⊢ (𝜑 → (𝑅1‘dom 𝑧) ∈ V) | |
10 | aomclem3.b | . . . 4 ⊢ 𝐵 = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐 ∈ (𝑅1‘∪ dom 𝑧)((𝑐 ∈ 𝑏 ∧ ¬ 𝑐 ∈ 𝑎) ∧ ∀𝑑 ∈ (𝑅1‘∪ dom 𝑧)(𝑑(𝑧‘∪ dom 𝑧)𝑐 → (𝑑 ∈ 𝑎 ↔ 𝑑 ∈ 𝑏)))} | |
11 | aomclem3.c | . . . 4 ⊢ 𝐶 = (𝑎 ∈ V ↦ sup((𝑦‘𝑎), (𝑅1‘dom 𝑧), 𝐵)) | |
12 | aomclem3.on | . . . 4 ⊢ (𝜑 → dom 𝑧 ∈ On) | |
13 | aomclem3.su | . . . 4 ⊢ (𝜑 → dom 𝑧 = suc ∪ dom 𝑧) | |
14 | aomclem3.we | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ dom 𝑧(𝑧‘𝑎) We (𝑅1‘𝑎)) | |
15 | aomclem3.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ On) | |
16 | aomclem3.za | . . . 4 ⊢ (𝜑 → dom 𝑧 ⊆ 𝐴) | |
17 | aomclem3.y | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘𝐴)(𝑎 ≠ ∅ → (𝑦‘𝑎) ∈ ((𝒫 𝑎 ∩ Fin) ∖ {∅}))) | |
18 | 10, 11, 12, 13, 14, 15, 16, 17 | aomclem2 41411 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎)) |
19 | neeq1 3007 | . . . . 5 ⊢ (𝑎 = 𝑑 → (𝑎 ≠ ∅ ↔ 𝑑 ≠ ∅)) | |
20 | fveq2 6847 | . . . . . 6 ⊢ (𝑎 = 𝑑 → (𝐶‘𝑎) = (𝐶‘𝑑)) | |
21 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑) | |
22 | 20, 21 | eleq12d 2832 | . . . . 5 ⊢ (𝑎 = 𝑑 → ((𝐶‘𝑎) ∈ 𝑎 ↔ (𝐶‘𝑑) ∈ 𝑑)) |
23 | 19, 22 | imbi12d 345 | . . . 4 ⊢ (𝑎 = 𝑑 → ((𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ (𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑))) |
24 | 23 | cbvralvw 3228 | . . 3 ⊢ (∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑)) |
25 | 18, 24 | sylib 217 | . 2 ⊢ (𝜑 → ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑)) |
26 | aomclem3.e | . 2 ⊢ 𝐸 = {⟨𝑎, 𝑏⟩ ∣ ∩ (◡𝐷 “ {𝑎}) ∈ ∩ (◡𝐷 “ {𝑏})} | |
27 | 8, 9, 25, 26 | dnwech 41404 | 1 ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∀wral 3065 ∃wrex 3074 Vcvv 3448 ∖ cdif 3912 ∩ cin 3914 ⊆ wss 3915 ∅c0 4287 𝒫 cpw 4565 {csn 4591 ∪ cuni 4870 ∩ cint 4912 class class class wbr 5110 {copab 5172 ↦ cmpt 5193 We wwe 5592 ◡ccnv 5637 dom cdm 5638 ran crn 5639 “ cima 5641 Oncon0 6322 suc csuc 6324 ‘cfv 6501 recscrecs 8321 Fincfn 8890 supcsup 9383 𝑅1cr1 9705 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-isom 6510 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-2o 8418 df-map 8774 df-en 8891 df-fin 8894 df-sup 9385 df-r1 9707 |
This theorem is referenced by: aomclem5 41414 |
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