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| Mirrors > Home > MPE Home > Th. List > Mathboxes > satff | Structured version Visualization version GIF version | ||
| Description: The satisfaction predicate as function over wff codes in the model 𝑀 and the binary relation 𝐸 on 𝑀. (Contributed by AV, 28-Oct-2023.) |
| Ref | Expression |
|---|---|
| satff | ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑁):(Fmla‘𝑁)⟶𝒫 (𝑀 ↑m ω)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | satffun 35414 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)) | |
| 2 | satfdmfmla 35405 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁)) | |
| 3 | df-fn 6564 | . . 3 ⊢ (((𝑀 Sat 𝐸)‘𝑁) Fn (Fmla‘𝑁) ↔ (Fun ((𝑀 Sat 𝐸)‘𝑁) ∧ dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁))) | |
| 4 | 1, 2, 3 | sylanbrc 583 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑁) Fn (Fmla‘𝑁)) |
| 5 | satfrnmapom 35375 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀 ↑m ω)) | |
| 6 | df-f 6565 | . 2 ⊢ (((𝑀 Sat 𝐸)‘𝑁):(Fmla‘𝑁)⟶𝒫 (𝑀 ↑m ω) ↔ (((𝑀 Sat 𝐸)‘𝑁) Fn (Fmla‘𝑁) ∧ ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀 ↑m ω))) | |
| 7 | 4, 5, 6 | sylanbrc 583 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑁):(Fmla‘𝑁)⟶𝒫 (𝑀 ↑m ω)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 𝒫 cpw 4600 dom cdm 5685 ran crn 5686 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ωcom 7887 ↑m cmap 8866 Sat csat 35341 Fmlacfmla 35342 |
| 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-nel 3047 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-int 4947 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-1o 8506 df-2o 8507 df-map 8868 df-goel 35345 df-gona 35346 df-goal 35347 df-sat 35348 df-fmla 35350 |
| This theorem is referenced by: satfun 35416 |
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