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Theorem fsovcnvlem 43067
Description: The 𝑂 operator, which maps between maps from one base set to subsets of the second to maps from the second base set to subsets of the first for base sets, gives a family of functions that include their own inverse. (Contributed by RP, 27-Apr-2021.)
Hypotheses
Ref Expression
fsovd.fs 𝑂 = (π‘Ž ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m π‘Ž) ↦ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ π‘Ž ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})))
fsovd.a (πœ‘ β†’ 𝐴 ∈ 𝑉)
fsovd.b (πœ‘ β†’ 𝐡 ∈ π‘Š)
fsovfvd.g 𝐺 = (𝐴𝑂𝐡)
fsovcnvlem.h 𝐻 = (𝐡𝑂𝐴)
Assertion
Ref Expression
fsovcnvlem (πœ‘ β†’ (𝐻 ∘ 𝐺) = ( I β†Ύ (𝒫 𝐡 ↑m 𝐴)))
Distinct variable groups:   𝐴,π‘Ž,𝑏,𝑓,π‘₯,𝑦   𝐡,π‘Ž,𝑏,𝑓,𝑦   πœ‘,π‘Ž,𝑏,𝑓,𝑦
Allowed substitution hints:   πœ‘(π‘₯)   𝐡(π‘₯)   𝐺(π‘₯,𝑦,𝑓,π‘Ž,𝑏)   𝐻(π‘₯,𝑦,𝑓,π‘Ž,𝑏)   𝑂(π‘₯,𝑦,𝑓,π‘Ž,𝑏)   𝑉(π‘₯,𝑦,𝑓,π‘Ž,𝑏)   π‘Š(π‘₯,𝑦,𝑓,π‘Ž,𝑏)

Proof of Theorem fsovcnvlem
Dummy variables 𝑐 𝑑 𝑔 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fsovd.a . . . . . . . 8 (πœ‘ β†’ 𝐴 ∈ 𝑉)
2 ssrab2 4078 . . . . . . . . 9 {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} βŠ† 𝐴
32a1i 11 . . . . . . . 8 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} βŠ† 𝐴)
41, 3sselpwd 5327 . . . . . . 7 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} ∈ 𝒫 𝐴)
54adantr 480 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐡) β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} ∈ 𝒫 𝐴)
65fmpttd 7117 . . . . 5 (πœ‘ β†’ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}):π΅βŸΆπ’« 𝐴)
71pwexd 5378 . . . . . 6 (πœ‘ β†’ 𝒫 𝐴 ∈ V)
8 fsovd.b . . . . . 6 (πœ‘ β†’ 𝐡 ∈ π‘Š)
97, 8elmapd 8837 . . . . 5 (πœ‘ β†’ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) ∈ (𝒫 𝐴 ↑m 𝐡) ↔ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}):π΅βŸΆπ’« 𝐴))
106, 9mpbird 256 . . . 4 (πœ‘ β†’ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) ∈ (𝒫 𝐴 ↑m 𝐡))
1110adantr 480 . . 3 ((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) β†’ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) ∈ (𝒫 𝐴 ↑m 𝐡))
12 fsovfvd.g . . . 4 𝐺 = (𝐴𝑂𝐡)
13 fsovd.fs . . . . 5 𝑂 = (π‘Ž ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m π‘Ž) ↦ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ π‘Ž ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})))
1413, 1, 8fsovd 43062 . . . 4 (πœ‘ β†’ (𝐴𝑂𝐡) = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})))
1512, 14eqtrid 2783 . . 3 (πœ‘ β†’ 𝐺 = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})))
16 fsovcnvlem.h . . . 4 𝐻 = (𝐡𝑂𝐴)
17 oveq2 7420 . . . . . . . 8 (π‘Ž = 𝑑 β†’ (𝒫 𝑏 ↑m π‘Ž) = (𝒫 𝑏 ↑m 𝑑))
18 rabeq 3445 . . . . . . . . 9 (π‘Ž = 𝑑 β†’ {π‘₯ ∈ π‘Ž ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} = {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})
1918mpteq2dv 5251 . . . . . . . 8 (π‘Ž = 𝑑 β†’ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ π‘Ž ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) = (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}))
2017, 19mpteq12dv 5240 . . . . . . 7 (π‘Ž = 𝑑 β†’ (𝑓 ∈ (𝒫 𝑏 ↑m π‘Ž) ↦ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ π‘Ž ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})) = (𝑓 ∈ (𝒫 𝑏 ↑m 𝑑) ↦ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})))
21 pweq 4617 . . . . . . . . 9 (𝑏 = 𝑐 β†’ 𝒫 𝑏 = 𝒫 𝑐)
2221oveq1d 7427 . . . . . . . 8 (𝑏 = 𝑐 β†’ (𝒫 𝑏 ↑m 𝑑) = (𝒫 𝑐 ↑m 𝑑))
23 mpteq1 5242 . . . . . . . 8 (𝑏 = 𝑐 β†’ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) = (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}))
2422, 23mpteq12dv 5240 . . . . . . 7 (𝑏 = 𝑐 β†’ (𝑓 ∈ (𝒫 𝑏 ↑m 𝑑) ↦ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})) = (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})))
2520, 24cbvmpov 7507 . . . . . 6 (π‘Ž ∈ V, 𝑏 ∈ V ↦ (𝑓 ∈ (𝒫 𝑏 ↑m π‘Ž) ↦ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ π‘Ž ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}))) = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})))
26 eqid 2731 . . . . . . 7 V = V
27 fveq1 6891 . . . . . . . . . . . 12 (𝑓 = 𝑔 β†’ (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯))
2827eleq2d 2818 . . . . . . . . . . 11 (𝑓 = 𝑔 β†’ (𝑦 ∈ (π‘“β€˜π‘₯) ↔ 𝑦 ∈ (π‘”β€˜π‘₯)))
2928rabbidv 3439 . . . . . . . . . 10 (𝑓 = 𝑔 β†’ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} = {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)})
3029mpteq2dv 5251 . . . . . . . . 9 (𝑓 = 𝑔 β†’ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) = (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)}))
3130cbvmptv 5262 . . . . . . . 8 (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})) = (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)}))
32 eleq1w 2815 . . . . . . . . . . . 12 (𝑦 = 𝑒 β†’ (𝑦 ∈ (π‘”β€˜π‘₯) ↔ 𝑒 ∈ (π‘”β€˜π‘₯)))
3332rabbidv 3439 . . . . . . . . . . 11 (𝑦 = 𝑒 β†’ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)} = {π‘₯ ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘₯)})
3433cbvmptv 5262 . . . . . . . . . 10 (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)}) = (𝑒 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘₯)})
35 fveq2 6892 . . . . . . . . . . . . 13 (π‘₯ = 𝑣 β†’ (π‘”β€˜π‘₯) = (π‘”β€˜π‘£))
3635eleq2d 2818 . . . . . . . . . . . 12 (π‘₯ = 𝑣 β†’ (𝑒 ∈ (π‘”β€˜π‘₯) ↔ 𝑒 ∈ (π‘”β€˜π‘£)))
3736cbvrabv 3441 . . . . . . . . . . 11 {π‘₯ ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘₯)} = {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)}
3837mpteq2i 5254 . . . . . . . . . 10 (𝑒 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘₯)}) = (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})
3934, 38eqtri 2759 . . . . . . . . 9 (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)}) = (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})
4039mpteq2i 5254 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)})) = (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)}))
4131, 40eqtri 2759 . . . . . . 7 (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})) = (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)}))
4226, 26, 41mpoeq123i 7488 . . . . . 6 (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑓 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}))) = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})))
4313, 25, 423eqtri 2763 . . . . 5 𝑂 = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})))
4443, 8, 1fsovd 43062 . . . 4 (πœ‘ β†’ (𝐡𝑂𝐴) = (𝑔 ∈ (𝒫 𝐴 ↑m 𝐡) ↦ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})))
4516, 44eqtrid 2783 . . 3 (πœ‘ β†’ 𝐻 = (𝑔 ∈ (𝒫 𝐴 ↑m 𝐡) ↦ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})))
46 fveq1 6891 . . . . . 6 (𝑔 = (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) β†’ (π‘”β€˜π‘£) = ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£))
4746eleq2d 2818 . . . . 5 (𝑔 = (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) β†’ (𝑒 ∈ (π‘”β€˜π‘£) ↔ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)))
4847rabbidv 3439 . . . 4 (𝑔 = (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) β†’ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ (π‘”β€˜π‘£)} = {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)})
4948mpteq2dv 5251 . . 3 (𝑔 = (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) β†’ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ (π‘”β€˜π‘£)}) = (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)}))
5011, 15, 45, 49fmptco 7130 . 2 (πœ‘ β†’ (𝐻 ∘ 𝐺) = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)})))
51 eqidd 2732 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) = (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}))
52 eleq1w 2815 . . . . . . . . . . . . 13 (𝑦 = 𝑣 β†’ (𝑦 ∈ (π‘“β€˜π‘₯) ↔ 𝑣 ∈ (π‘“β€˜π‘₯)))
5352rabbidv 3439 . . . . . . . . . . . 12 (𝑦 = 𝑣 β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)})
5453adantl 481 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) ∧ 𝑦 = 𝑣) β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)})
55 simpr 484 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ 𝑣 ∈ 𝐡)
56 rabexg 5332 . . . . . . . . . . . . 13 (𝐴 ∈ 𝑉 β†’ {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)} ∈ V)
571, 56syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)} ∈ V)
5857ad2antrr 723 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)} ∈ V)
5951, 54, 55, 58fvmptd 7006 . . . . . . . . . 10 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£) = {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)})
6059eleq2d 2818 . . . . . . . . 9 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ (𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£) ↔ 𝑒 ∈ {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)}))
61 fveq2 6892 . . . . . . . . . . . 12 (π‘₯ = 𝑒 β†’ (π‘“β€˜π‘₯) = (π‘“β€˜π‘’))
6261eleq2d 2818 . . . . . . . . . . 11 (π‘₯ = 𝑒 β†’ (𝑣 ∈ (π‘“β€˜π‘₯) ↔ 𝑣 ∈ (π‘“β€˜π‘’)))
6362elrab3 3685 . . . . . . . . . 10 (𝑒 ∈ 𝐴 β†’ (𝑒 ∈ {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)} ↔ 𝑣 ∈ (π‘“β€˜π‘’)))
6463ad2antlr 724 . . . . . . . . 9 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ (𝑒 ∈ {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)} ↔ 𝑣 ∈ (π‘“β€˜π‘’)))
6560, 64bitrd 278 . . . . . . . 8 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ (𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£) ↔ 𝑣 ∈ (π‘“β€˜π‘’)))
6665rabbidva 3438 . . . . . . 7 ((πœ‘ ∧ 𝑒 ∈ 𝐴) β†’ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)} = {𝑣 ∈ 𝐡 ∣ 𝑣 ∈ (π‘“β€˜π‘’)})
6766adantlr 712 . . . . . 6 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)} = {𝑣 ∈ 𝐡 ∣ 𝑣 ∈ (π‘“β€˜π‘’)})
68 elmapi 8846 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) β†’ 𝑓:π΄βŸΆπ’« 𝐡)
6968ad2antlr 724 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑓:π΄βŸΆπ’« 𝐡)
70 simpr 484 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ 𝐴)
7169, 70ffvelcdmd 7088 . . . . . . . . 9 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ (π‘“β€˜π‘’) ∈ 𝒫 𝐡)
7271elpwid 4612 . . . . . . . 8 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ (π‘“β€˜π‘’) βŠ† 𝐡)
73 sseqin2 4216 . . . . . . . 8 ((π‘“β€˜π‘’) βŠ† 𝐡 ↔ (𝐡 ∩ (π‘“β€˜π‘’)) = (π‘“β€˜π‘’))
7472, 73sylib 217 . . . . . . 7 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ (𝐡 ∩ (π‘“β€˜π‘’)) = (π‘“β€˜π‘’))
75 dfin5 3957 . . . . . . 7 (𝐡 ∩ (π‘“β€˜π‘’)) = {𝑣 ∈ 𝐡 ∣ 𝑣 ∈ (π‘“β€˜π‘’)}
7674, 75eqtr3di 2786 . . . . . 6 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ (π‘“β€˜π‘’) = {𝑣 ∈ 𝐡 ∣ 𝑣 ∈ (π‘“β€˜π‘’)})
7767, 76eqtr4d 2774 . . . . 5 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)} = (π‘“β€˜π‘’))
7877mpteq2dva 5249 . . . 4 ((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) β†’ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)}) = (𝑒 ∈ 𝐴 ↦ (π‘“β€˜π‘’)))
7968feqmptd 6961 . . . . 5 (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) β†’ 𝑓 = (𝑒 ∈ 𝐴 ↦ (π‘“β€˜π‘’)))
8079adantl 481 . . . 4 ((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) β†’ 𝑓 = (𝑒 ∈ 𝐴 ↦ (π‘“β€˜π‘’)))
8178, 80eqtr4d 2774 . . 3 ((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) β†’ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)}) = 𝑓)
8281mpteq2dva 5249 . 2 (πœ‘ β†’ (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ ((𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})β€˜π‘£)})) = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ 𝑓))
83 mptresid 6051 . . . 4 ( I β†Ύ (𝒫 𝐡 ↑m 𝐴)) = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ 𝑓)
8483eqcomi 2740 . . 3 (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ 𝑓) = ( I β†Ύ (𝒫 𝐡 ↑m 𝐴))
8584a1i 11 . 2 (πœ‘ β†’ (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ 𝑓) = ( I β†Ύ (𝒫 𝐡 ↑m 𝐴)))
8650, 82, 853eqtrd 2775 1 (πœ‘ β†’ (𝐻 ∘ 𝐺) = ( I β†Ύ (𝒫 𝐡 ↑m 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {crab 3431  Vcvv 3473   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603   ↦ cmpt 5232   I cid 5574   β†Ύ cres 5679   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7412   ∈ cmpo 7414   ↑m cmap 8823
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7978  df-2nd 7979  df-map 8825
This theorem is referenced by:  fsovcnvd  43068
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