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| Mirrors > Home > MPE Home > Th. List > Mathboxes > satf0sucom | Structured version Visualization version GIF version | ||
| Description: The satisfaction predicate as function over wff codes in the empty model with an empty binary relation at a successor of ω. (Contributed by AV, 14-Sep-2023.) |
| Ref | Expression |
|---|---|
| satf0sucom | ⊢ (𝑁 ∈ suc ω → ((∅ Sat ∅)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑁)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | satf0 35377 | . . 3 ⊢ (∅ Sat ∅) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}) ↾ suc ω) | |
| 2 | 1 | fveq1i 6907 | . 2 ⊢ ((∅ Sat ∅)‘𝑁) = ((rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}) ↾ suc ω)‘𝑁) |
| 3 | fvres 6925 | . 2 ⊢ (𝑁 ∈ suc ω → ((rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))}) ↾ suc ω)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑁)) | |
| 4 | 2, 3 | eqtrid 2789 | 1 ⊢ (𝑁 ∈ suc ω → ((∅ Sat ∅)‘𝑁) = (rec((𝑓 ∈ V ↦ (𝑓 ∪ {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑢 ∈ 𝑓 (∃𝑣 ∈ 𝑓 𝑥 = ((1st ‘𝑢)⊼𝑔(1st ‘𝑣)) ∨ ∃𝑖 ∈ ω 𝑥 = ∀𝑔𝑖(1st ‘𝑢)))})), {〈𝑥, 𝑦〉 ∣ (𝑦 = ∅ ∧ ∃𝑖 ∈ ω ∃𝑗 ∈ ω 𝑥 = (𝑖∈𝑔𝑗))})‘𝑁)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 Vcvv 3480 ∪ cun 3949 ∅c0 4333 {copab 5205 ↦ cmpt 5225 ↾ cres 5687 suc csuc 6386 ‘cfv 6561 (class class class)co 7431 ωcom 7887 1st c1st 8012 reccrdg 8449 ∈𝑔cgoe 35338 ⊼𝑔cgna 35339 ∀𝑔cgol 35340 Sat csat 35341 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-map 8868 df-sat 35348 |
| This theorem is referenced by: satf00 35379 satf0suc 35381 sat1el2xp 35384 fmla 35386 fmlasuc0 35389 |
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