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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  satfv0fvfmla0 Structured version   Visualization version   GIF version

Theorem satfv0fvfmla0 34702
Description: The value of the satisfaction predicate as function over a wff code at βˆ…. (Contributed by AV, 2-Nov-2023.)
Hypothesis
Ref Expression
satfv0fv.s 𝑆 = (𝑀 Sat 𝐸)
Assertion
Ref Expression
satfv0fvfmla0 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ ((π‘†β€˜βˆ…)β€˜π‘‹) = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))})
Distinct variable groups:   𝐸,π‘Ž   𝑀,π‘Ž   𝑋,π‘Ž
Allowed substitution hints:   𝑆(π‘Ž)   𝑉(π‘Ž)   π‘Š(π‘Ž)

Proof of Theorem satfv0fvfmla0
Dummy variables 𝑖 𝑗 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 satfv0fun 34660 . . . 4 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ Fun ((𝑀 Sat 𝐸)β€˜βˆ…))
2 satfv0fv.s . . . . . 6 𝑆 = (𝑀 Sat 𝐸)
32fveq1i 6891 . . . . 5 (π‘†β€˜βˆ…) = ((𝑀 Sat 𝐸)β€˜βˆ…)
43funeqi 6568 . . . 4 (Fun (π‘†β€˜βˆ…) ↔ Fun ((𝑀 Sat 𝐸)β€˜βˆ…))
51, 4sylibr 233 . . 3 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ Fun (π‘†β€˜βˆ…))
653adant3 1130 . 2 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ Fun (π‘†β€˜βˆ…))
7 fmla0 34671 . . . . . . . 8 (Fmlaβ€˜βˆ…) = {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)}
87eleq2i 2823 . . . . . . 7 (𝑋 ∈ (Fmlaβ€˜βˆ…) ↔ 𝑋 ∈ {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)})
9 eqeq1 2734 . . . . . . . . 9 (π‘₯ = 𝑋 β†’ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ↔ 𝑋 = (π‘–βˆˆπ‘”π‘—)))
1092rexbidv 3217 . . . . . . . 8 (π‘₯ = 𝑋 β†’ (βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—) ↔ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝑋 = (π‘–βˆˆπ‘”π‘—)))
1110elrab 3682 . . . . . . 7 (𝑋 ∈ {π‘₯ ∈ V ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ π‘₯ = (π‘–βˆˆπ‘”π‘—)} ↔ (𝑋 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝑋 = (π‘–βˆˆπ‘”π‘—)))
128, 11bitri 274 . . . . . 6 (𝑋 ∈ (Fmlaβ€˜βˆ…) ↔ (𝑋 ∈ V ∧ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝑋 = (π‘–βˆˆπ‘”π‘—)))
13 simpr 483 . . . . . . . . . 10 (((𝑖 ∈ Ο‰ ∧ 𝑗 ∈ Ο‰) ∧ 𝑋 = (π‘–βˆˆπ‘”π‘—)) β†’ 𝑋 = (π‘–βˆˆπ‘”π‘—))
14 goel 34636 . . . . . . . . . . . . . 14 ((𝑖 ∈ Ο‰ ∧ 𝑗 ∈ Ο‰) β†’ (π‘–βˆˆπ‘”π‘—) = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©)
1514eqeq2d 2741 . . . . . . . . . . . . 13 ((𝑖 ∈ Ο‰ ∧ 𝑗 ∈ Ο‰) β†’ (𝑋 = (π‘–βˆˆπ‘”π‘—) ↔ 𝑋 = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©))
16 2fveq3 6895 . . . . . . . . . . . . . . . 16 (𝑋 = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ© β†’ (1st β€˜(2nd β€˜π‘‹)) = (1st β€˜(2nd β€˜βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©)))
17 0ex 5306 . . . . . . . . . . . . . . . . . . 19 βˆ… ∈ V
18 opex 5463 . . . . . . . . . . . . . . . . . . 19 βŸ¨π‘–, π‘—βŸ© ∈ V
1917, 18op2nd 7986 . . . . . . . . . . . . . . . . . 18 (2nd β€˜βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©) = βŸ¨π‘–, π‘—βŸ©
2019fveq2i 6893 . . . . . . . . . . . . . . . . 17 (1st β€˜(2nd β€˜βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©)) = (1st β€˜βŸ¨π‘–, π‘—βŸ©)
21 vex 3476 . . . . . . . . . . . . . . . . . 18 𝑖 ∈ V
22 vex 3476 . . . . . . . . . . . . . . . . . 18 𝑗 ∈ V
2321, 22op1st 7985 . . . . . . . . . . . . . . . . 17 (1st β€˜βŸ¨π‘–, π‘—βŸ©) = 𝑖
2420, 23eqtri 2758 . . . . . . . . . . . . . . . 16 (1st β€˜(2nd β€˜βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©)) = 𝑖
2516, 24eqtrdi 2786 . . . . . . . . . . . . . . 15 (𝑋 = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ© β†’ (1st β€˜(2nd β€˜π‘‹)) = 𝑖)
2625fveq2d 6894 . . . . . . . . . . . . . 14 (𝑋 = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ© β†’ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹))) = (π‘Žβ€˜π‘–))
27 2fveq3 6895 . . . . . . . . . . . . . . . 16 (𝑋 = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ© β†’ (2nd β€˜(2nd β€˜π‘‹)) = (2nd β€˜(2nd β€˜βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©)))
2819fveq2i 6893 . . . . . . . . . . . . . . . . 17 (2nd β€˜(2nd β€˜βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©)) = (2nd β€˜βŸ¨π‘–, π‘—βŸ©)
2921, 22op2nd 7986 . . . . . . . . . . . . . . . . 17 (2nd β€˜βŸ¨π‘–, π‘—βŸ©) = 𝑗
3028, 29eqtri 2758 . . . . . . . . . . . . . . . 16 (2nd β€˜(2nd β€˜βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ©)) = 𝑗
3127, 30eqtrdi 2786 . . . . . . . . . . . . . . 15 (𝑋 = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ© β†’ (2nd β€˜(2nd β€˜π‘‹)) = 𝑗)
3231fveq2d 6894 . . . . . . . . . . . . . 14 (𝑋 = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ© β†’ (π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹))) = (π‘Žβ€˜π‘—))
3326, 32breq12d 5160 . . . . . . . . . . . . 13 (𝑋 = βŸ¨βˆ…, βŸ¨π‘–, π‘—βŸ©βŸ© β†’ ((π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹))) ↔ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)))
3415, 33syl6bi 252 . . . . . . . . . . . 12 ((𝑖 ∈ Ο‰ ∧ 𝑗 ∈ Ο‰) β†’ (𝑋 = (π‘–βˆˆπ‘”π‘—) β†’ ((π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹))) ↔ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—))))
3534imp 405 . . . . . . . . . . 11 (((𝑖 ∈ Ο‰ ∧ 𝑗 ∈ Ο‰) ∧ 𝑋 = (π‘–βˆˆπ‘”π‘—)) β†’ ((π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹))) ↔ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)))
3635rabbidv 3438 . . . . . . . . . 10 (((𝑖 ∈ Ο‰ ∧ 𝑗 ∈ Ο‰) ∧ 𝑋 = (π‘–βˆˆπ‘”π‘—)) β†’ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})
3713, 36jca 510 . . . . . . . . 9 (((𝑖 ∈ Ο‰ ∧ 𝑗 ∈ Ο‰) ∧ 𝑋 = (π‘–βˆˆπ‘”π‘—)) β†’ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)}))
3837ex 411 . . . . . . . 8 ((𝑖 ∈ Ο‰ ∧ 𝑗 ∈ Ο‰) β†’ (𝑋 = (π‘–βˆˆπ‘”π‘—) β†’ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})))
3938reximdva 3166 . . . . . . 7 (𝑖 ∈ Ο‰ β†’ (βˆƒπ‘— ∈ Ο‰ 𝑋 = (π‘–βˆˆπ‘”π‘—) β†’ βˆƒπ‘— ∈ Ο‰ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})))
4039reximia 3079 . . . . . 6 (βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ 𝑋 = (π‘–βˆˆπ‘”π‘—) β†’ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)}))
4112, 40simplbiim 503 . . . . 5 (𝑋 ∈ (Fmlaβ€˜βˆ…) β†’ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)}))
42413ad2ant3 1133 . . . 4 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)}))
43 simp3 1136 . . . . 5 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ 𝑋 ∈ (Fmlaβ€˜βˆ…))
44 ovex 7444 . . . . . 6 (𝑀 ↑m Ο‰) ∈ V
4544rabex 5331 . . . . 5 {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} ∈ V
46 eqeq1 2734 . . . . . . . 8 (𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} β†’ (𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)} ↔ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)}))
479, 46bi2anan9 635 . . . . . . 7 ((π‘₯ = 𝑋 ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}) β†’ ((π‘₯ = (π‘–βˆˆπ‘”π‘—) ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)}) ↔ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})))
48472rexbidv 3217 . . . . . 6 ((π‘₯ = 𝑋 ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}) β†’ (βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)}) ↔ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})))
4948opelopabga 5532 . . . . 5 ((𝑋 ∈ (Fmlaβ€˜βˆ…) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} ∈ V) β†’ (βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})} ↔ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})))
5043, 45, 49sylancl 584 . . . 4 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ (βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})} ↔ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (𝑋 = (π‘–βˆˆπ‘”π‘—) ∧ {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))} = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})))
5142, 50mpbird 256 . . 3 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})})
522satfv0 34647 . . . . 5 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ (π‘†β€˜βˆ…) = {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})})
5352eleq2d 2817 . . . 4 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š) β†’ (βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ (π‘†β€˜βˆ…) ↔ βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})}))
54533adant3 1130 . . 3 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ (βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ (π‘†β€˜βˆ…) ↔ βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ {⟨π‘₯, π‘¦βŸ© ∣ βˆƒπ‘– ∈ Ο‰ βˆƒπ‘— ∈ Ο‰ (π‘₯ = (π‘–βˆˆπ‘”π‘—) ∧ 𝑦 = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜π‘–)𝐸(π‘Žβ€˜π‘—)})}))
5551, 54mpbird 256 . 2 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ (π‘†β€˜βˆ…))
56 funopfv 6942 . 2 (Fun (π‘†β€˜βˆ…) β†’ (βŸ¨π‘‹, {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}⟩ ∈ (π‘†β€˜βˆ…) β†’ ((π‘†β€˜βˆ…)β€˜π‘‹) = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))}))
576, 55, 56sylc 65 1 ((𝑀 ∈ 𝑉 ∧ 𝐸 ∈ π‘Š ∧ 𝑋 ∈ (Fmlaβ€˜βˆ…)) β†’ ((π‘†β€˜βˆ…)β€˜π‘‹) = {π‘Ž ∈ (𝑀 ↑m Ο‰) ∣ (π‘Žβ€˜(1st β€˜(2nd β€˜π‘‹)))𝐸(π‘Žβ€˜(2nd β€˜(2nd β€˜π‘‹)))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068  {crab 3430  Vcvv 3472  βˆ…c0 4321  βŸ¨cop 4633   class class class wbr 5147  {copab 5209  Fun wfun 6536  β€˜cfv 6542  (class class class)co 7411  Ο‰com 7857  1st c1st 7975  2nd c2nd 7976   ↑m cmap 8822  βˆˆπ‘”cgoe 34622   Sat csat 34625  Fmlacfmla 34626
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 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 34629  df-sat 34632  df-fmla 34634
This theorem is referenced by:  satefvfmla0  34707
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