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

Theorem ssmapsn 42260
 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 3894 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵m {𝐴}))
3 elmapi 8444 . . . . . . . 8 (𝑓 ∈ (𝐵m {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 6504 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . . 8 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovexd 7191 . . . . . . . . . . 11 (𝜑 → (𝐵m {𝐴}) ∈ V)
98, 1ssexd 5198 . . . . . . . . . 10 (𝜑𝐶 ∈ V)
10 rnexg 7620 . . . . . . . . . . . 12 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 3080 . . . . . . . . . . 11 𝑓𝐶 ran 𝑓 ∈ V
1211a1i 11 . . . . . . . . . 10 (𝜑 → ∀𝑓𝐶 ran 𝑓 ∈ V)
139, 12jca 515 . . . . . . . . 9 (𝜑 → (𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V))
14 iunexg 7674 . . . . . . . . 9 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
1513, 14syl 17 . . . . . . . 8 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
167, 15eqeltrd 2852 . . . . . . 7 (𝜑𝐷 ∈ V)
1716adantr 484 . . . . . 6 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
18 ssiun2 4939 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1918adantl 485 . . . . . . . 8 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
20 ssmapsn.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
21 snidg 4559 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2322adantr 484 . . . . . . . . 9 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
24 fnfvelrn 6845 . . . . . . . . 9 ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓𝐴) ∈ ran 𝑓)
255, 23, 24syl2anc 587 . . . . . . . 8 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2619, 25sseldd 3895 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2726, 6eleqtrrdi 2863 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
285, 17, 27elmapsnd 42248 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷m {𝐴}))
2928ex 416 . . . 4 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
3016adantr 484 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐷 ∈ V)
31 snex 5304 . . . . . . . . . 10 {𝐴} ∈ V
3231a1i 11 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → {𝐴} ∈ V)
33 simpr 488 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝑓 ∈ (𝐷m {𝐴}))
3422adantr 484 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐴 ∈ {𝐴})
3530, 32, 33, 34fvmap 42241 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝐷)
366idi 1 . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
37 rneq 5782 . . . . . . . . . 10 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3837cbviunv 4932 . . . . . . . . 9 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
3936, 38eqtri 2781 . . . . . . . 8 𝐷 = 𝑔𝐶 ran 𝑔
4035, 39eleqtrdi 2862 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
41 eliun 4890 . . . . . . 7 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
4240, 41sylib 221 . . . . . 6 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
43 simp3 1135 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
44 simp1l 1194 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4544, 20syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
46 eqid 2758 . . . . . . . . . . 11 {𝐴} = {𝐴}
47 simp1r 1195 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷m {𝐴}))
48 elmapfn 8460 . . . . . . . . . . . 12 (𝑓 ∈ (𝐷m {𝐴}) → 𝑓 Fn {𝐴})
4947, 48syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
501sselda 3894 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵m {𝐴}))
51 elmapfn 8460 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐵m {𝐴}) → 𝑔 Fn {𝐴})
5250, 51syl 17 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
53523adant3 1129 . . . . . . . . . . . 12 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
54533adant1r 1174 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5545, 46, 49, 54fsneqrn 42255 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓𝐴) ∈ ran 𝑔))
5643, 55mpbird 260 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
57 simp2 1134 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5856, 57eqeltrd 2852 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
59583exp 1116 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑔𝐶 → ((𝑓𝐴) ∈ ran 𝑔𝑓𝐶)))
6059rexlimdv 3207 . . . . . 6 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
6142, 60mpd 15 . . . . 5 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝑓𝐶)
6261ex 416 . . . 4 (𝜑 → (𝑓 ∈ (𝐷m {𝐴}) → 𝑓𝐶))
6329, 62impbid 215 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
6463alrimiv 1928 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
65 nfcv 2919 . . 3 𝑓𝐶
66 ssmapsn.f . . . 4 𝑓𝐷
67 nfcv 2919 . . . 4 𝑓m
68 nfcv 2919 . . . 4 𝑓{𝐴}
6966, 67, 68nfov 7186 . . 3 𝑓(𝐷m {𝐴})
7065, 69cleqf 2947 . 2 (𝐶 = (𝐷m {𝐴}) ↔ ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
7164, 70sylibr 237 1 (𝜑𝐶 = (𝐷m {𝐴}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2899  ∀wral 3070  ∃wrex 3071  Vcvv 3409   ⊆ wss 3860  {csn 4525  ∪ ciun 4886  ran crn 5529   Fn wfn 6335  ⟶wf 6336  ‘cfv 6340  (class class class)co 7156   ↑m cmap 8422 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-map 8424 This theorem is referenced by:  vonvolmbllem  43710  vonvolmbl2  43713  vonvol2  43714
 Copyright terms: Public domain W3C validator