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Mirrors > Home > MPE Home > Th. List > Mathboxes > satefv | Structured version Visualization version GIF version |
Description: The simplified satisfaction predicate as function over wff codes in the model 𝑀 at the code 𝑈. (Contributed by AV, 30-Oct-2023.) |
Ref | Expression |
---|---|
satefv | ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀 Sat∈ 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-sate 32612 | . . 3 ⊢ Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢)) | |
2 | 1 | a1i 11 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → Sat∈ = (𝑚 ∈ V, 𝑢 ∈ V ↦ (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢))) |
3 | id 22 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → 𝑚 = 𝑀) | |
4 | 3 | sqxpeqd 5580 | . . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑚 × 𝑚) = (𝑀 × 𝑀)) |
5 | 4 | ineq2d 4182 | . . . . . . 7 ⊢ (𝑚 = 𝑀 → ( E ∩ (𝑚 × 𝑚)) = ( E ∩ (𝑀 × 𝑀))) |
6 | 3, 5 | oveq12d 7167 | . . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑚 Sat ( E ∩ (𝑚 × 𝑚))) = (𝑀 Sat ( E ∩ (𝑀 × 𝑀)))) |
7 | 6 | fveq1d 6665 | . . . . 5 ⊢ (𝑚 = 𝑀 → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)) |
8 | 7 | adantr 483 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → ((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω) = ((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)) |
9 | simpr 487 | . . . 4 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → 𝑢 = 𝑈) | |
10 | 8, 9 | fveq12d 6670 | . . 3 ⊢ ((𝑚 = 𝑀 ∧ 𝑢 = 𝑈) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈)) |
11 | 10 | adantl 484 | . 2 ⊢ (((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) ∧ (𝑚 = 𝑀 ∧ 𝑢 = 𝑈)) → (((𝑚 Sat ( E ∩ (𝑚 × 𝑚)))‘ω)‘𝑢) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈)) |
12 | elex 3509 | . . 3 ⊢ (𝑀 ∈ 𝑉 → 𝑀 ∈ V) | |
13 | 12 | adantr 483 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → 𝑀 ∈ V) |
14 | elex 3509 | . . 3 ⊢ (𝑈 ∈ 𝑊 → 𝑈 ∈ V) | |
15 | 14 | adantl 484 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → 𝑈 ∈ V) |
16 | fvexd 6678 | . 2 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈) ∈ V) | |
17 | 2, 11, 13, 15, 16 | ovmpod 7295 | 1 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑈 ∈ 𝑊) → (𝑀 Sat∈ 𝑈) = (((𝑀 Sat ( E ∩ (𝑀 × 𝑀)))‘ω)‘𝑈)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∩ cin 3928 E cep 5457 × cxp 5546 ‘cfv 6348 (class class class)co 7149 ∈ cmpo 7151 ωcom 7573 Sat csat 32604 Sat∈ csate 32606 |
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-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 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-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7152 df-oprab 7153 df-mpo 7154 df-sate 32612 |
This theorem is referenced by: sate0 32683 satef 32684 satefvfmla0 32686 satefvfmla1 32693 |
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