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Theorem fmlafvel 34931
Description: A class is a valid Godel formula of height 𝑁 iff it is the first component of a member of the value of the satisfaction predicate as function over wff codes in the empty model with an empty binary relation at 𝑁. (Contributed by AV, 19-Sep-2023.)
Assertion
Ref Expression
fmlafvel (𝑁 ∈ Ο‰ β†’ (𝐹 ∈ (Fmlaβ€˜π‘) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘)))

Proof of Theorem fmlafvel
Dummy variables 𝑒 𝑣 π‘₯ 𝑦 𝑖 𝑗 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6891 . . . . . . 7 (π‘₯ = βˆ… β†’ (Fmlaβ€˜π‘₯) = (Fmlaβ€˜βˆ…))
21eleq2d 2814 . . . . . 6 (π‘₯ = βˆ… β†’ (𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ 𝐹 ∈ (Fmlaβ€˜βˆ…)))
3 fveq2 6891 . . . . . . 7 (π‘₯ = βˆ… β†’ ((βˆ… Sat βˆ…)β€˜π‘₯) = ((βˆ… Sat βˆ…)β€˜βˆ…))
43eleq2d 2814 . . . . . 6 (π‘₯ = βˆ… β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜βˆ…)))
52, 4bibi12d 345 . . . . 5 (π‘₯ = βˆ… β†’ ((𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯)) ↔ (𝐹 ∈ (Fmlaβ€˜βˆ…) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜βˆ…))))
65imbi2d 340 . . . 4 (π‘₯ = βˆ… β†’ ((𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯))) ↔ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜βˆ…) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜βˆ…)))))
7 fveq2 6891 . . . . . . 7 (π‘₯ = 𝑦 β†’ (Fmlaβ€˜π‘₯) = (Fmlaβ€˜π‘¦))
87eleq2d 2814 . . . . . 6 (π‘₯ = 𝑦 β†’ (𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ 𝐹 ∈ (Fmlaβ€˜π‘¦)))
9 fveq2 6891 . . . . . . 7 (π‘₯ = 𝑦 β†’ ((βˆ… Sat βˆ…)β€˜π‘₯) = ((βˆ… Sat βˆ…)β€˜π‘¦))
109eleq2d 2814 . . . . . 6 (π‘₯ = 𝑦 β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))
118, 10bibi12d 345 . . . . 5 (π‘₯ = 𝑦 β†’ ((𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯)) ↔ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦))))
1211imbi2d 340 . . . 4 (π‘₯ = 𝑦 β†’ ((𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯))) ↔ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))))
13 fveq2 6891 . . . . . . 7 (π‘₯ = suc 𝑦 β†’ (Fmlaβ€˜π‘₯) = (Fmlaβ€˜suc 𝑦))
1413eleq2d 2814 . . . . . 6 (π‘₯ = suc 𝑦 β†’ (𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ 𝐹 ∈ (Fmlaβ€˜suc 𝑦)))
15 fveq2 6891 . . . . . . 7 (π‘₯ = suc 𝑦 β†’ ((βˆ… Sat βˆ…)β€˜π‘₯) = ((βˆ… Sat βˆ…)β€˜suc 𝑦))
1615eleq2d 2814 . . . . . 6 (π‘₯ = suc 𝑦 β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜suc 𝑦)))
1714, 16bibi12d 345 . . . . 5 (π‘₯ = suc 𝑦 β†’ ((𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯)) ↔ (𝐹 ∈ (Fmlaβ€˜suc 𝑦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜suc 𝑦))))
1817imbi2d 340 . . . 4 (π‘₯ = suc 𝑦 β†’ ((𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯))) ↔ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜suc 𝑦)))))
19 fveq2 6891 . . . . . . 7 (π‘₯ = 𝑁 β†’ (Fmlaβ€˜π‘₯) = (Fmlaβ€˜π‘))
2019eleq2d 2814 . . . . . 6 (π‘₯ = 𝑁 β†’ (𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ 𝐹 ∈ (Fmlaβ€˜π‘)))
21 fveq2 6891 . . . . . . 7 (π‘₯ = 𝑁 β†’ ((βˆ… Sat βˆ…)β€˜π‘₯) = ((βˆ… Sat βˆ…)β€˜π‘))
2221eleq2d 2814 . . . . . 6 (π‘₯ = 𝑁 β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘)))
2320, 22bibi12d 345 . . . . 5 (π‘₯ = 𝑁 β†’ ((𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯)) ↔ (𝐹 ∈ (Fmlaβ€˜π‘) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘))))
2423imbi2d 340 . . . 4 (π‘₯ = 𝑁 β†’ ((𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘₯) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘₯))) ↔ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘)))))
25 eqeq1 2731 . . . . . . . 8 (π‘₯ = 𝐹 β†’ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ↔ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
26252rexbidv 3214 . . . . . . 7 (π‘₯ = 𝐹 β†’ (βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—) ↔ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
2726elrab 3680 . . . . . 6 (𝐹 ∈ {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)} ↔ (𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
28 eqidd 2728 . . . . . . . 8 ((𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ βˆ… = βˆ…)
29 simpr 484 . . . . . . . 8 ((𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))
3028, 29jca 511 . . . . . . 7 ((𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
31 simpr 484 . . . . . . . . 9 ((βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))
3231anim2i 616 . . . . . . . 8 ((𝐹 ∈ V ∧ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))) β†’ (𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
3332ex 412 . . . . . . 7 (𝐹 ∈ V β†’ ((βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ (𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))))
3430, 33impbid2 225 . . . . . 6 (𝐹 ∈ V β†’ ((𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) ↔ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))))
3527, 34bitrid 283 . . . . 5 (𝐹 ∈ V β†’ (𝐹 ∈ {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)} ↔ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))))
36 fmla0 34928 . . . . . . 7 (Fmlaβ€˜βˆ…) = {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)}
3736eleq2i 2820 . . . . . 6 (𝐹 ∈ (Fmlaβ€˜βˆ…) ↔ 𝐹 ∈ {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)})
3837a1i 11 . . . . 5 (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜βˆ…) ↔ 𝐹 ∈ {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)}))
39 satf00 34920 . . . . . . . 8 ((βˆ… Sat βˆ…)β€˜βˆ…) = {⟨π‘₯, π‘¦βŸ© ∣ (𝑦 = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—))}
4039a1i 11 . . . . . . 7 (𝐹 ∈ V β†’ ((βˆ… Sat βˆ…)β€˜βˆ…) = {⟨π‘₯, π‘¦βŸ© ∣ (𝑦 = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—))})
4140eleq2d 2814 . . . . . 6 (𝐹 ∈ V β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜βˆ…) ↔ ⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ (𝑦 = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—))}))
42 0ex 5301 . . . . . . 7 βˆ… ∈ V
43 eqeq1 2731 . . . . . . . . 9 (𝑦 = βˆ… β†’ (𝑦 = βˆ… ↔ βˆ… = βˆ…))
4443, 26bi2anan9r 638 . . . . . . . 8 ((π‘₯ = 𝐹 ∧ 𝑦 = βˆ…) β†’ ((𝑦 = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)) ↔ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))))
4544opelopabga 5529 . . . . . . 7 ((𝐹 ∈ V ∧ βˆ… ∈ V) β†’ (⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ (𝑦 = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—))} ↔ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))))
4642, 45mpan2 690 . . . . . 6 (𝐹 ∈ V β†’ (⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘¦βŸ© ∣ (𝑦 = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—))} ↔ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))))
4741, 46bitrd 279 . . . . 5 (𝐹 ∈ V β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜βˆ…) ↔ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))))
4835, 38, 473bitr4d 311 . . . 4 (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜βˆ…) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜βˆ…)))
49 eqid 2727 . . . . . . . . . . . 12 βˆ… = βˆ…
5049biantrur 530 . . . . . . . . . . 11 (βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’)) ↔ (βˆ… = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’))))
5150bicomi 223 . . . . . . . . . 10 ((βˆ… = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’))) ↔ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’)))
5251a1i 11 . . . . . . . . 9 (𝐹 ∈ V β†’ ((βˆ… = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’))) ↔ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’))))
53 eqeq1 2731 . . . . . . . . . . . 12 (𝑧 = βˆ… β†’ (𝑧 = βˆ… ↔ βˆ… = βˆ…))
54 eqeq1 2731 . . . . . . . . . . . . . . 15 (π‘₯ = 𝐹 β†’ (π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ↔ 𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£))))
5554rexbidv 3173 . . . . . . . . . . . . . 14 (π‘₯ = 𝐹 β†’ (βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ↔ βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£))))
56 eqeq1 2731 . . . . . . . . . . . . . . 15 (π‘₯ = 𝐹 β†’ (π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’) ↔ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’)))
5756rexbidv 3173 . . . . . . . . . . . . . 14 (π‘₯ = 𝐹 β†’ (βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’) ↔ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’)))
5855, 57orbi12d 917 . . . . . . . . . . . . 13 (π‘₯ = 𝐹 β†’ ((βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)) ↔ (βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’))))
5958rexbidv 3173 . . . . . . . . . . . 12 (π‘₯ = 𝐹 β†’ (βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)) ↔ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’))))
6053, 59bi2anan9r 638 . . . . . . . . . . 11 ((π‘₯ = 𝐹 ∧ 𝑧 = βˆ…) β†’ ((𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))) ↔ (βˆ… = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’)))))
6160opelopabga 5529 . . . . . . . . . 10 ((𝐹 ∈ V ∧ βˆ… ∈ V) β†’ (⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))} ↔ (βˆ… = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’)))))
6242, 61mpan2 690 . . . . . . . . 9 (𝐹 ∈ V β†’ (⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))} ↔ (βˆ… = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’)))))
6359elabg 3663 . . . . . . . . 9 (𝐹 ∈ V β†’ (𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))} ↔ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’))))
6452, 62, 633bitr4d 311 . . . . . . . 8 (𝐹 ∈ V β†’ (⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))} ↔ 𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))}))
6564adantl 481 . . . . . . 7 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ (⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))} ↔ 𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))}))
6665orbi2d 914 . . . . . 6 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ ((⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦) ∨ ⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))}) ↔ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦) ∨ 𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))})))
67 eqid 2727 . . . . . . . . . 10 (βˆ… Sat βˆ…) = (βˆ… Sat βˆ…)
6867satf0suc 34922 . . . . . . . . 9 (𝑦 ∈ Ο‰ β†’ ((βˆ… Sat βˆ…)β€˜suc 𝑦) = (((βˆ… Sat βˆ…)β€˜π‘¦) βˆͺ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))}))
6968eleq2d 2814 . . . . . . . 8 (𝑦 ∈ Ο‰ β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜suc 𝑦) ↔ ⟨𝐹, βˆ…βŸ© ∈ (((βˆ… Sat βˆ…)β€˜π‘¦) βˆͺ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))})))
70 elun 4144 . . . . . . . 8 (⟨𝐹, βˆ…βŸ© ∈ (((βˆ… Sat βˆ…)β€˜π‘¦) βˆͺ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))}) ↔ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦) ∨ ⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))}))
7169, 70bitrdi 287 . . . . . . 7 (𝑦 ∈ Ο‰ β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜suc 𝑦) ↔ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦) ∨ ⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))})))
7271ad2antrr 725 . . . . . 6 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜suc 𝑦) ↔ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦) ∨ ⟨𝐹, βˆ…βŸ© ∈ {⟨π‘₯, π‘§βŸ© ∣ (𝑧 = βˆ… ∧ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’)))})))
73 fmlasuc0 34930 . . . . . . . . 9 (𝑦 ∈ Ο‰ β†’ (Fmlaβ€˜suc 𝑦) = ((Fmlaβ€˜π‘¦) βˆͺ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))}))
7473eleq2d 2814 . . . . . . . 8 (𝑦 ∈ Ο‰ β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑦) ↔ 𝐹 ∈ ((Fmlaβ€˜π‘¦) βˆͺ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))})))
7574ad2antrr 725 . . . . . . 7 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑦) ↔ 𝐹 ∈ ((Fmlaβ€˜π‘¦) βˆͺ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))})))
76 elun 4144 . . . . . . . 8 (𝐹 ∈ ((Fmlaβ€˜π‘¦) βˆͺ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))}) ↔ (𝐹 ∈ (Fmlaβ€˜π‘¦) ∨ 𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))}))
7776a1i 11 . . . . . . 7 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ (𝐹 ∈ ((Fmlaβ€˜π‘¦) βˆͺ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))}) ↔ (𝐹 ∈ (Fmlaβ€˜π‘¦) ∨ 𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))})))
78 simpr 484 . . . . . . . . 9 ((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) β†’ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦))))
7978imp 406 . . . . . . . 8 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))
8079orbi1d 915 . . . . . . 7 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ ((𝐹 ∈ (Fmlaβ€˜π‘¦) ∨ 𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))}) ↔ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦) ∨ 𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))})))
8175, 77, 803bitrd 305 . . . . . 6 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑦) ↔ (⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦) ∨ 𝐹 ∈ {π‘₯ ∣ βˆƒπ‘’ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)(βˆƒπ‘£ ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ∨ βˆƒπ‘– ∈ Ο‰ π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’))})))
8266, 72, 813bitr4rd 312 . . . . 5 (((𝑦 ∈ Ο‰ ∧ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦)))) ∧ 𝐹 ∈ V) β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜suc 𝑦)))
8382exp31 419 . . . 4 (𝑦 ∈ Ο‰ β†’ ((𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘¦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘¦))) β†’ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜suc 𝑦) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜suc 𝑦)))))
846, 12, 18, 24, 48, 83finds 7898 . . 3 (𝑁 ∈ Ο‰ β†’ (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜π‘) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘))))
8584com12 32 . 2 (𝐹 ∈ V β†’ (𝑁 ∈ Ο‰ β†’ (𝐹 ∈ (Fmlaβ€˜π‘) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘))))
86 prcnel 3493 . . . . 5 (Β¬ 𝐹 ∈ V β†’ Β¬ 𝐹 ∈ (Fmlaβ€˜π‘))
8786adantr 480 . . . 4 ((Β¬ 𝐹 ∈ V ∧ 𝑁 ∈ Ο‰) β†’ Β¬ 𝐹 ∈ (Fmlaβ€˜π‘))
88 opprc1 4893 . . . . . 6 (Β¬ 𝐹 ∈ V β†’ ⟨𝐹, βˆ…βŸ© = βˆ…)
8988adantr 480 . . . . 5 ((Β¬ 𝐹 ∈ V ∧ 𝑁 ∈ Ο‰) β†’ ⟨𝐹, βˆ…βŸ© = βˆ…)
90 satf0n0 34924 . . . . . . 7 (𝑁 ∈ Ο‰ β†’ βˆ… βˆ‰ ((βˆ… Sat βˆ…)β€˜π‘))
91 df-nel 3042 . . . . . . 7 (βˆ… βˆ‰ ((βˆ… Sat βˆ…)β€˜π‘) ↔ Β¬ βˆ… ∈ ((βˆ… Sat βˆ…)β€˜π‘))
9290, 91sylib 217 . . . . . 6 (𝑁 ∈ Ο‰ β†’ Β¬ βˆ… ∈ ((βˆ… Sat βˆ…)β€˜π‘))
9392adantl 481 . . . . 5 ((Β¬ 𝐹 ∈ V ∧ 𝑁 ∈ Ο‰) β†’ Β¬ βˆ… ∈ ((βˆ… Sat βˆ…)β€˜π‘))
9489, 93eqneltrd 2848 . . . 4 ((Β¬ 𝐹 ∈ V ∧ 𝑁 ∈ Ο‰) β†’ Β¬ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘))
9587, 942falsed 376 . . 3 ((Β¬ 𝐹 ∈ V ∧ 𝑁 ∈ Ο‰) β†’ (𝐹 ∈ (Fmlaβ€˜π‘) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘)))
9695ex 412 . 2 (Β¬ 𝐹 ∈ V β†’ (𝑁 ∈ Ο‰ β†’ (𝐹 ∈ (Fmlaβ€˜π‘) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘))))
9785, 96pm2.61i 182 1 (𝑁 ∈ Ο‰ β†’ (𝐹 ∈ (Fmlaβ€˜π‘) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜π‘)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099  {cab 2704   βˆ‰ wnel 3041  βˆƒwrex 3065  {crab 3427  Vcvv 3469   βˆͺ cun 3942  βˆ…c0 4318  βŸ¨cop 4630  {copab 5204  suc csuc 6365  β€˜cfv 6542  (class class class)co 7414  Ο‰com 7864  1st c1st 7985  βˆˆπ‘”cgoe 34879  βŠΌπ‘”cgna 34880  βˆ€π‘”cgol 34881   Sat csat 34882  Fmlacfmla 34883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9656
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-map 8838  df-goel 34886  df-sat 34889  df-fmla 34891
This theorem is referenced by:  fmlasuc  34932
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