Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ssmapsn Structured version   Visualization version   GIF version

Theorem ssmapsn 43915
Description: A subset 𝐶 of a set exponentiation to a singleton, is its projection 𝐷 exponentiated to the singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
ssmapsn.f 𝑓𝐷
ssmapsn.a (𝜑𝐴𝑉)
ssmapsn.c (𝜑𝐶 ⊆ (𝐵m {𝐴}))
ssmapsn.d 𝐷 = 𝑓𝐶 ran 𝑓
Assertion
Ref Expression
ssmapsn (𝜑𝐶 = (𝐷m {𝐴}))
Distinct variable groups:   𝐴,𝑓   𝐶,𝑓   𝜑,𝑓
Allowed substitution hints:   𝐵(𝑓)   𝐷(𝑓)   𝑉(𝑓)

Proof of Theorem ssmapsn
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 ssmapsn.c . . . . . . . . 9 (𝜑𝐶 ⊆ (𝐵m {𝐴}))
21sselda 3983 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵m {𝐴}))
3 elmapi 8843 . . . . . . . 8 (𝑓 ∈ (𝐵m {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 6719 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . . 8 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovexd 7444 . . . . . . . . . . 11 (𝜑 → (𝐵m {𝐴}) ∈ V)
98, 1ssexd 5325 . . . . . . . . . 10 (𝜑𝐶 ∈ V)
10 rnexg 7895 . . . . . . . . . . . 12 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 3064 . . . . . . . . . . 11 𝑓𝐶 ran 𝑓 ∈ V
1211a1i 11 . . . . . . . . . 10 (𝜑 → ∀𝑓𝐶 ran 𝑓 ∈ V)
139, 12jca 513 . . . . . . . . 9 (𝜑 → (𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V))
14 iunexg 7950 . . . . . . . . 9 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
1513, 14syl 17 . . . . . . . 8 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
167, 15eqeltrd 2834 . . . . . . 7 (𝜑𝐷 ∈ V)
1716adantr 482 . . . . . 6 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
18 ssiun2 5051 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1918adantl 483 . . . . . . . 8 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
20 ssmapsn.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
21 snidg 4663 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2322adantr 482 . . . . . . . . 9 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
24 fnfvelrn 7083 . . . . . . . . 9 ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓𝐴) ∈ ran 𝑓)
255, 23, 24syl2anc 585 . . . . . . . 8 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2619, 25sseldd 3984 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2726, 6eleqtrrdi 2845 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
285, 17, 27elmapsnd 43903 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷m {𝐴}))
2928ex 414 . . . 4 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
3016adantr 482 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐷 ∈ V)
31 snex 5432 . . . . . . . . . 10 {𝐴} ∈ V
3231a1i 11 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → {𝐴} ∈ V)
33 simpr 486 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝑓 ∈ (𝐷m {𝐴}))
3422adantr 482 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐴 ∈ {𝐴})
3530, 32, 33, 34fvmap 43897 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝐷)
366idi 1 . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
37 rneq 5936 . . . . . . . . . 10 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3837cbviunv 5044 . . . . . . . . 9 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
3936, 38eqtri 2761 . . . . . . . 8 𝐷 = 𝑔𝐶 ran 𝑔
4035, 39eleqtrdi 2844 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
41 eliun 5002 . . . . . . 7 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
4240, 41sylib 217 . . . . . 6 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
43 simp3 1139 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
44 simp1l 1198 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4544, 20syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
46 eqid 2733 . . . . . . . . . . 11 {𝐴} = {𝐴}
47 simp1r 1199 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷m {𝐴}))
48 elmapfn 8859 . . . . . . . . . . . 12 (𝑓 ∈ (𝐷m {𝐴}) → 𝑓 Fn {𝐴})
4947, 48syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
501sselda 3983 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵m {𝐴}))
51 elmapfn 8859 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐵m {𝐴}) → 𝑔 Fn {𝐴})
5250, 51syl 17 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
53523adant3 1133 . . . . . . . . . . . 12 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
54533adant1r 1178 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5545, 46, 49, 54fsneqrn 43910 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓𝐴) ∈ ran 𝑔))
5643, 55mpbird 257 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
57 simp2 1138 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5856, 57eqeltrd 2834 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
59583exp 1120 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑔𝐶 → ((𝑓𝐴) ∈ ran 𝑔𝑓𝐶)))
6059rexlimdv 3154 . . . . . 6 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
6142, 60mpd 15 . . . . 5 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝑓𝐶)
6261ex 414 . . . 4 (𝜑 → (𝑓 ∈ (𝐷m {𝐴}) → 𝑓𝐶))
6329, 62impbid 211 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
6463alrimiv 1931 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
65 nfcv 2904 . . 3 𝑓𝐶
66 ssmapsn.f . . . 4 𝑓𝐷
67 nfcv 2904 . . . 4 𝑓m
68 nfcv 2904 . . . 4 𝑓{𝐴}
6966, 67, 68nfov 7439 . . 3 𝑓(𝐷m {𝐴})
7065, 69cleqf 2935 . 2 (𝐶 = (𝐷m {𝐴}) ↔ ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
7164, 70sylibr 233 1 (𝜑𝐶 = (𝐷m {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088  wal 1540   = wceq 1542  wcel 2107  wnfc 2884  wral 3062  wrex 3071  Vcvv 3475  wss 3949  {csn 4629   ciun 4998  ran crn 5678   Fn wfn 6539  wf 6540  cfv 6544  (class class class)co 7409  m cmap 8820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  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 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-map 8822
This theorem is referenced by:  vonvolmbllem  45376  vonvolmbl2  45379  vonvol2  45380
  Copyright terms: Public domain W3C validator