Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
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 32675 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → Fun ((𝑀 Sat 𝐸)‘𝑁)) | |
2 | satfdmfmla 32666 | . . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → dom ((𝑀 Sat 𝐸)‘𝑁) = (Fmla‘𝑁)) | |
3 | df-fn 6351 | . . 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 585 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ((𝑀 Sat 𝐸)‘𝑁) Fn (Fmla‘𝑁)) |
5 | satfrnmapom 32636 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ 𝑊 ∧ 𝑁 ∈ ω) → ran ((𝑀 Sat 𝐸)‘𝑁) ⊆ 𝒫 (𝑀 ↑m ω)) | |
6 | df-f 6352 | . 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 585 | 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 1082 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 𝒫 cpw 4532 dom cdm 5548 ran crn 5549 Fun wfun 6342 Fn wfn 6343 ⟶wf 6344 ‘cfv 6348 (class class class)co 7149 ωcom 7573 ↑m cmap 8399 Sat csat 32602 Fmlacfmla 32603 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-inf2 9097 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-2o 8096 df-map 8401 df-goel 32606 df-gona 32607 df-goal 32608 df-sat 32609 df-fmla 32611 |
This theorem is referenced by: satfun 32677 |
Copyright terms: Public domain | W3C validator |