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Mirrors > Home > MPE Home > Th. List > Mathboxes > aomclem3 | Structured version Visualization version GIF version |
Description: Lemma for dfac11 39655. 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 5800 | . . . . . . 7 ⊢ (𝑎 = 𝑐 → ran 𝑎 = ran 𝑐) | |
3 | 2 | difeq2d 4098 | . . . . . 6 ⊢ (𝑎 = 𝑐 → ((𝑅1‘dom 𝑧) ∖ ran 𝑎) = ((𝑅1‘dom 𝑧) ∖ ran 𝑐)) |
4 | 3 | fveq2d 6668 | . . . . 5 ⊢ (𝑎 = 𝑐 → (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎)) = (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))) |
5 | 4 | cbvmptv 5161 | . . . 4 ⊢ (𝑎 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑎))) = (𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐))) |
6 | recseq 8004 | . . . 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 2844 | . 2 ⊢ 𝐷 = recs((𝑐 ∈ V ↦ (𝐶‘((𝑅1‘dom 𝑧) ∖ ran 𝑐)))) |
9 | fvexd 6679 | . 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 39648 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎)) |
19 | neeq1 3078 | . . . . 5 ⊢ (𝑎 = 𝑑 → (𝑎 ≠ ∅ ↔ 𝑑 ≠ ∅)) | |
20 | fveq2 6664 | . . . . . 6 ⊢ (𝑎 = 𝑑 → (𝐶‘𝑎) = (𝐶‘𝑑)) | |
21 | id 22 | . . . . . 6 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑) | |
22 | 20, 21 | eleq12d 2907 | . . . . 5 ⊢ (𝑎 = 𝑑 → ((𝐶‘𝑎) ∈ 𝑎 ↔ (𝐶‘𝑑) ∈ 𝑑)) |
23 | 19, 22 | imbi12d 347 | . . . 4 ⊢ (𝑎 = 𝑑 → ((𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ (𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑))) |
24 | 23 | cbvralvw 3449 | . . 3 ⊢ (∀𝑎 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑎 ≠ ∅ → (𝐶‘𝑎) ∈ 𝑎) ↔ ∀𝑑 ∈ 𝒫 (𝑅1‘dom 𝑧)(𝑑 ≠ ∅ → (𝐶‘𝑑) ∈ 𝑑)) |
25 | 18, 24 | sylib 220 | . 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 39641 | 1 ⊢ (𝜑 → 𝐸 We (𝑅1‘dom 𝑧)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 {csn 4560 ∪ cuni 4831 ∩ cint 4868 class class class wbr 5058 {copab 5120 ↦ cmpt 5138 We wwe 5507 ◡ccnv 5548 dom cdm 5549 ran crn 5550 “ cima 5552 Oncon0 6185 suc csuc 6187 ‘cfv 6349 recscrecs 8001 Fincfn 8503 supcsup 8898 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 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 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-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-er 8283 df-map 8402 df-en 8504 df-fin 8507 df-sup 8900 df-r1 9187 |
This theorem is referenced by: aomclem5 39651 |
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