Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fsovcnvlem Structured version   Visualization version   GIF version

Theorem fsovcnvlem 42846
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 4077 . . . . . . . . 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 5326 . . . . . . 7 (πœ‘ β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} ∈ 𝒫 𝐴)
54adantr 481 . . . . . 6 ((πœ‘ ∧ 𝑦 ∈ 𝐡) β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} ∈ 𝒫 𝐴)
65fmpttd 7116 . . . . 5 (πœ‘ β†’ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}):π΅βŸΆπ’« 𝐴)
71pwexd 5377 . . . . . 6 (πœ‘ β†’ 𝒫 𝐴 ∈ V)
8 fsovd.b . . . . . 6 (πœ‘ β†’ 𝐡 ∈ π‘Š)
97, 8elmapd 8836 . . . . 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 481 . . 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 42841 . . . 4 (πœ‘ β†’ (𝐴𝑂𝐡) = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ (𝑦 ∈ 𝐡 ↦ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})))
1512, 14eqtrid 2784 . . 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 7419 . . . . . . . 8 (π‘Ž = 𝑑 β†’ (𝒫 𝑏 ↑m π‘Ž) = (𝒫 𝑏 ↑m 𝑑))
18 rabeq 3446 . . . . . . . . 9 (π‘Ž = 𝑑 β†’ {π‘₯ ∈ π‘Ž ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} = {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)})
1918mpteq2dv 5250 . . . . . . . 8 (π‘Ž = 𝑑 β†’ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ π‘Ž ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) = (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}))
2017, 19mpteq12dv 5239 . . . . . . 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 4616 . . . . . . . . 9 (𝑏 = 𝑐 β†’ 𝒫 𝑏 = 𝒫 𝑐)
2221oveq1d 7426 . . . . . . . 8 (𝑏 = 𝑐 β†’ (𝒫 𝑏 ↑m 𝑑) = (𝒫 𝑐 ↑m 𝑑))
23 mpteq1 5241 . . . . . . . 8 (𝑏 = 𝑐 β†’ (𝑦 ∈ 𝑏 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) = (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}))
2422, 23mpteq12dv 5239 . . . . . . 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 7506 . . . . . 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 2732 . . . . . . 7 V = V
27 fveq1 6890 . . . . . . . . . . . 12 (𝑓 = 𝑔 β†’ (π‘“β€˜π‘₯) = (π‘”β€˜π‘₯))
2827eleq2d 2819 . . . . . . . . . . 11 (𝑓 = 𝑔 β†’ (𝑦 ∈ (π‘“β€˜π‘₯) ↔ 𝑦 ∈ (π‘”β€˜π‘₯)))
2928rabbidv 3440 . . . . . . . . . 10 (𝑓 = 𝑔 β†’ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} = {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)})
3029mpteq2dv 5250 . . . . . . . . 9 (𝑓 = 𝑔 β†’ (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)}) = (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)}))
3130cbvmptv 5261 . . . . . . . 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 2816 . . . . . . . . . . . 12 (𝑦 = 𝑒 β†’ (𝑦 ∈ (π‘”β€˜π‘₯) ↔ 𝑒 ∈ (π‘”β€˜π‘₯)))
3332rabbidv 3440 . . . . . . . . . . 11 (𝑦 = 𝑒 β†’ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)} = {π‘₯ ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘₯)})
3433cbvmptv 5261 . . . . . . . . . 10 (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)}) = (𝑒 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘₯)})
35 fveq2 6891 . . . . . . . . . . . . 13 (π‘₯ = 𝑣 β†’ (π‘”β€˜π‘₯) = (π‘”β€˜π‘£))
3635eleq2d 2819 . . . . . . . . . . . 12 (π‘₯ = 𝑣 β†’ (𝑒 ∈ (π‘”β€˜π‘₯) ↔ 𝑒 ∈ (π‘”β€˜π‘£)))
3736cbvrabv 3442 . . . . . . . . . . 11 {π‘₯ ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘₯)} = {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)}
3837mpteq2i 5253 . . . . . . . . . 10 (𝑒 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘₯)}) = (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})
3934, 38eqtri 2760 . . . . . . . . 9 (𝑦 ∈ 𝑐 ↦ {π‘₯ ∈ 𝑑 ∣ 𝑦 ∈ (π‘”β€˜π‘₯)}) = (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})
4039mpteq2i 5253 . . . . . . . 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 2760 . . . . . . 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 7487 . . . . . 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 2764 . . . . 5 𝑂 = (𝑑 ∈ V, 𝑐 ∈ V ↦ (𝑔 ∈ (𝒫 𝑐 ↑m 𝑑) ↦ (𝑒 ∈ 𝑐 ↦ {𝑣 ∈ 𝑑 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})))
4443, 8, 1fsovd 42841 . . . 4 (πœ‘ β†’ (𝐡𝑂𝐴) = (𝑔 ∈ (𝒫 𝐴 ↑m 𝐡) ↦ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})))
4516, 44eqtrid 2784 . . 3 (πœ‘ β†’ 𝐻 = (𝑔 ∈ (𝒫 𝐴 ↑m 𝐡) ↦ (𝑒 ∈ 𝐴 ↦ {𝑣 ∈ 𝐡 ∣ 𝑒 ∈ (π‘”β€˜π‘£)})))
46 fveq1 6890 . . . . . 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 2819 . . . . 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 3440 . . . 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 5250 . . 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 7129 . 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 2733 . . . . . . . . . . 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 2816 . . . . . . . . . . . . 13 (𝑦 = 𝑣 β†’ (𝑦 ∈ (π‘“β€˜π‘₯) ↔ 𝑣 ∈ (π‘“β€˜π‘₯)))
5352rabbidv 3440 . . . . . . . . . . . 12 (𝑦 = 𝑣 β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)})
5453adantl 482 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) ∧ 𝑦 = 𝑣) β†’ {π‘₯ ∈ 𝐴 ∣ 𝑦 ∈ (π‘“β€˜π‘₯)} = {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)})
55 simpr 485 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ 𝑣 ∈ 𝐡)
56 rabexg 5331 . . . . . . . . . . . . 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 724 . . . . . . . . . . 11 (((πœ‘ ∧ 𝑒 ∈ 𝐴) ∧ 𝑣 ∈ 𝐡) β†’ {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)} ∈ V)
5951, 54, 55, 58fvmptd 7005 . . . . . . . . . 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 2819 . . . . . . . . 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 6891 . . . . . . . . . . . 12 (π‘₯ = 𝑒 β†’ (π‘“β€˜π‘₯) = (π‘“β€˜π‘’))
6261eleq2d 2819 . . . . . . . . . . 11 (π‘₯ = 𝑒 β†’ (𝑣 ∈ (π‘“β€˜π‘₯) ↔ 𝑣 ∈ (π‘“β€˜π‘’)))
6362elrab3 3684 . . . . . . . . . 10 (𝑒 ∈ 𝐴 β†’ (𝑒 ∈ {π‘₯ ∈ 𝐴 ∣ 𝑣 ∈ (π‘“β€˜π‘₯)} ↔ 𝑣 ∈ (π‘“β€˜π‘’)))
6463ad2antlr 725 . . . . . . . . 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 3439 . . . . . . 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 713 . . . . . 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 8845 . . . . . . . . . . 11 (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) β†’ 𝑓:π΄βŸΆπ’« 𝐡)
6968ad2antlr 725 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑓:π΄βŸΆπ’« 𝐡)
70 simpr 485 . . . . . . . . . 10 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ 𝑒 ∈ 𝐴)
7169, 70ffvelcdmd 7087 . . . . . . . . 9 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ (π‘“β€˜π‘’) ∈ 𝒫 𝐡)
7271elpwid 4611 . . . . . . . 8 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ (π‘“β€˜π‘’) βŠ† 𝐡)
73 sseqin2 4215 . . . . . . . 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 3956 . . . . . . 7 (𝐡 ∩ (π‘“β€˜π‘’)) = {𝑣 ∈ 𝐡 ∣ 𝑣 ∈ (π‘“β€˜π‘’)}
7674, 75eqtr3di 2787 . . . . . 6 (((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) ∧ 𝑒 ∈ 𝐴) β†’ (π‘“β€˜π‘’) = {𝑣 ∈ 𝐡 ∣ 𝑣 ∈ (π‘“β€˜π‘’)})
7767, 76eqtr4d 2775 . . . . 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 5248 . . . 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 6960 . . . . 5 (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) β†’ 𝑓 = (𝑒 ∈ 𝐴 ↦ (π‘“β€˜π‘’)))
8079adantl 482 . . . 4 ((πœ‘ ∧ 𝑓 ∈ (𝒫 𝐡 ↑m 𝐴)) β†’ 𝑓 = (𝑒 ∈ 𝐴 ↦ (π‘“β€˜π‘’)))
8178, 80eqtr4d 2775 . . 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 5248 . 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 6050 . . . 4 ( I β†Ύ (𝒫 𝐡 ↑m 𝐴)) = (𝑓 ∈ (𝒫 𝐡 ↑m 𝐴) ↦ 𝑓)
8483eqcomi 2741 . . 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 2776 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 396   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602   ↦ cmpt 5231   I cid 5573   β†Ύ cres 5678   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7411   ∈ cmpo 7413   ↑m cmap 8822
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
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  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-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-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-id 5574  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-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-1st 7977  df-2nd 7978  df-map 8824
This theorem is referenced by:  fsovcnvd  42847
  Copyright terms: Public domain W3C validator