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| Mirrors > Home > MPE Home > Th. List > Mathboxes > satfn | Structured version Visualization version GIF version | ||
| Description: The satisfaction predicate for wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀 is a function over suc ω. (Contributed by AV, 6-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| satfn | ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) Fn suc ω) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rdgfnon 8459 | . . . 4 ⊢ rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) Fn On | |
| 2 | 1 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) Fn On) | 
| 3 | ordom 7898 | . . . . 5 ⊢ Ord ω | |
| 4 | ordsuc 7834 | . . . . 5 ⊢ (Ord ω ↔ Ord suc ω) | |
| 5 | 3, 4 | mpbi 230 | . . . 4 ⊢ Ord suc ω | 
| 6 | ordsson 7804 | . . . 4 ⊢ (Ord suc ω → suc ω ⊆ On) | |
| 7 | 5, 6 | mp1i 13 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → suc ω ⊆ On) | 
| 8 | 2, 7 | fnssresd 6691 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω) Fn suc ω) | 
| 9 | satf 35359 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω)) | |
| 10 | 9 | fneq1d 6660 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → ((𝑀 Sat 𝐸) Fn suc ω ↔ (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 (𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∧ 𝑦 = ((𝑀 ↑m ω) ∖ ((2nd ‘𝑢) ∩ (2nd ‘𝑣)))) ∨ ∃𝑖 ∈ ω (𝑥 = ∀𝑔𝑖(1st ‘𝑢) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ ∀𝑧 ∈ 𝑀 ({〈𝑖, 𝑧〉} ∪ (𝑎 ↾ (ω ∖ {𝑖}))) ∈ (2nd ‘𝑢)}))})), {〈𝑥, 𝑦〉 ∣ ∃𝑖 ∈ ω ∃𝑗 ∈ ω (𝑥 = (𝑖∈𝑔𝑗) ∧ 𝑦 = {𝑎 ∈ (𝑀 ↑m ω) ∣ (𝑎‘𝑖)𝐸(𝑎‘𝑗)})}) ↾ suc ω) Fn suc ω)) | 
| 11 | 8, 10 | mpbird 257 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊) → (𝑀 Sat 𝐸) Fn suc ω) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 {crab 3435 Vcvv 3479 ∖ cdif 3947 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 {csn 4625 〈cop 4631 class class class wbr 5142 {copab 5204 ↦ cmpt 5224 ↾ cres 5686 Ord word 6382 Oncon0 6383 suc csuc 6385 Fn wfn 6555 ‘cfv 6560 (class class class)co 7432 ωcom 7888 1st c1st 8013 2nd c2nd 8014 reccrdg 8450 ↑m cmap 8867 ∈𝑔cgoe 35339 ⊼𝑔cgna 35340 ∀𝑔cgol 35341 Sat csat 35342 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 ax-inf2 9682 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-sat 35349 | 
| This theorem is referenced by: (None) | 
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