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Theorem fmlafvel 34445
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 2819 . . . . . 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 2819 . . . . . 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 2819 . . . . . 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 2819 . . . . . 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 2819 . . . . . 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 2819 . . . . . 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 2819 . . . . . 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 2819 . . . . . 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 2736 . . . . . . . 8 (π‘₯ = 𝐹 β†’ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ↔ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
26252rexbidv 3219 . . . . . . 7 (π‘₯ = 𝐹 β†’ (βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—) ↔ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
2726elrab 3683 . . . . . 6 (𝐹 ∈ {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)} ↔ (𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
28 eqidd 2733 . . . . . . . 8 ((𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ βˆ… = βˆ…)
29 simpr 485 . . . . . . . 8 ((𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))
3028, 29jca 512 . . . . . . 7 ((𝐹 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ (βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)))
31 simpr 485 . . . . . . . . 9 ((βˆ… = βˆ… ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—)) β†’ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝐹 = (π‘–βˆˆπ‘”π‘—))
3231anim2i 617 . . . . . . . 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 413 . . . . . . 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 282 . . . . 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 34442 . . . . . . 7 (Fmlaβ€˜βˆ…) = {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)}
3736eleq2i 2825 . . . . . 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 34434 . . . . . . . 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 2819 . . . . . 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 5307 . . . . . . 7 βˆ… ∈ V
43 eqeq1 2736 . . . . . . . . 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 5533 . . . . . . 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 689 . . . . . 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 278 . . . . 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 310 . . . 4 (𝐹 ∈ V β†’ (𝐹 ∈ (Fmlaβ€˜βˆ…) ↔ ⟨𝐹, βˆ…βŸ© ∈ ((βˆ… Sat βˆ…)β€˜βˆ…)))
49 eqid 2732 . . . . . . . . . . . 12 βˆ… = βˆ…
5049biantrur 531 . . . . . . . . . . 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 2736 . . . . . . . . . . . 12 (𝑧 = βˆ… β†’ (𝑧 = βˆ… ↔ βˆ… = βˆ…))
54 eqeq1 2736 . . . . . . . . . . . . . . 15 (π‘₯ = 𝐹 β†’ (π‘₯ = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£)) ↔ 𝐹 = ((1st β€˜π‘’)βŠΌπ‘”(1st β€˜π‘£))))
5554rexbidv 3178 . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . 15 (π‘₯ = 𝐹 β†’ (π‘₯ = βˆ€π‘”π‘–(1st β€˜π‘’) ↔ 𝐹 = βˆ€π‘”π‘–(1st β€˜π‘’)))
5756rexbidv 3178 . . . . . . . . . . . . . 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 3178 . . . . . . . . . . . 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 5533 . . . . . . . . . 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 689 . . . . . . . . 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 3666 . . . . . . . . 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 310 . . . . . . . 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 482 . . . . . . 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 2732 . . . . . . . . . 10 (βˆ… Sat βˆ…) = (βˆ… Sat βˆ…)
6867satf0suc 34436 . . . . . . . . 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 2819 . . . . . . . 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 4148 . . . . . . . 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 286 . . . . . . 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 724 . . . . . 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 34444 . . . . . . . . 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 2819 . . . . . . . 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 724 . . . . . . 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 4148 . . . . . . . 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 485 . . . . . . . . 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 407 . . . . . . . 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 304 . . . . . 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 311 . . . . 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 420 . . . 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 7891 . . 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 3497 . . . . 5 (Β¬ 𝐹 ∈ V β†’ Β¬ 𝐹 ∈ (Fmlaβ€˜π‘))
8786adantr 481 . . . 4 ((Β¬ 𝐹 ∈ V ∧ 𝑁 ∈ Ο‰) β†’ Β¬ 𝐹 ∈ (Fmlaβ€˜π‘))
88 opprc1 4897 . . . . . 6 (Β¬ 𝐹 ∈ V β†’ ⟨𝐹, βˆ…βŸ© = βˆ…)
8988adantr 481 . . . . 5 ((Β¬ 𝐹 ∈ V ∧ 𝑁 ∈ Ο‰) β†’ ⟨𝐹, βˆ…βŸ© = βˆ…)
90 satf0n0 34438 . . . . . . 7 (𝑁 ∈ Ο‰ β†’ βˆ… βˆ‰ ((βˆ… Sat βˆ…)β€˜π‘))
91 df-nel 3047 . . . . . . 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 482 . . . . 5 ((Β¬ 𝐹 ∈ V ∧ 𝑁 ∈ Ο‰) β†’ Β¬ βˆ… ∈ ((βˆ… Sat βˆ…)β€˜π‘))
9489, 93eqneltrd 2853 . . . 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 413 . 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106  {cab 2709   βˆ‰ wnel 3046  βˆƒwrex 3070  {crab 3432  Vcvv 3474   βˆͺ cun 3946  βˆ…c0 4322  βŸ¨cop 4634  {copab 5210  suc csuc 6366  β€˜cfv 6543  (class class class)co 7411  Ο‰com 7857  1st c1st 7975  βˆˆπ‘”cgoe 34393  βŠΌπ‘”cgna 34394  βˆ€π‘”cgol 34395   Sat csat 34396  Fmlacfmla 34397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-map 8824  df-goel 34400  df-sat 34403  df-fmla 34405
This theorem is referenced by:  fmlasuc  34446
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