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Theorem ssmapsn 45221
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 . . . . . . . 8 (𝜑𝐶 ⊆ (𝐵m {𝐴}))
21sselda 3983 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵m {𝐴}))
3 elmapi 8889 . . . . . . 7 (𝑓 ∈ (𝐵m {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 6737 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . 8 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . 7 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovexd 7466 . . . . . . . . 9 (𝜑 → (𝐵m {𝐴}) ∈ V)
98, 1ssexd 5324 . . . . . . . 8 (𝜑𝐶 ∈ V)
10 rnexg 7924 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 3063 . . . . . . . 8 𝑓𝐶 ran 𝑓 ∈ V
12 iunexg 7988 . . . . . . . 8 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
139, 11, 12sylancl 586 . . . . . . 7 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
147, 13eqeltrd 2841 . . . . . 6 (𝜑𝐷 ∈ V)
1514adantr 480 . . . . 5 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
16 ssiun2 5047 . . . . . . . 8 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1716adantl 481 . . . . . . 7 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
18 ssmapsn.a . . . . . . . . . 10 (𝜑𝐴𝑉)
19 snidg 4660 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {𝐴})
2018, 19syl 17 . . . . . . . . 9 (𝜑𝐴 ∈ {𝐴})
2120adantr 480 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
225, 21fnfvelrnd 7102 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2317, 22sseldd 3984 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2423, 6eleqtrrdi 2852 . . . . 5 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
255, 15, 24elmapsnd 45209 . . . 4 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷m {𝐴}))
2614adantr 480 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐷 ∈ V)
27 snex 5436 . . . . . . . . 9 {𝐴} ∈ V
2827a1i 11 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → {𝐴} ∈ V)
29 simpr 484 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝑓 ∈ (𝐷m {𝐴}))
3020adantr 480 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐴 ∈ {𝐴})
3126, 28, 29, 30fvmap 45203 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝐷)
32 rneq 5947 . . . . . . . . 9 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3332cbviunv 5040 . . . . . . . 8 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
346, 33eqtri 2765 . . . . . . 7 𝐷 = 𝑔𝐶 ran 𝑔
3531, 34eleqtrdi 2851 . . . . . 6 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
36 eliun 4995 . . . . . 6 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
3735, 36sylib 218 . . . . 5 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
38 simp3 1139 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
39 simp1l 1198 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4039, 18syl 17 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
41 eqid 2737 . . . . . . . . 9 {𝐴} = {𝐴}
42 simp1r 1199 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷m {𝐴}))
43 elmapfn 8905 . . . . . . . . . 10 (𝑓 ∈ (𝐷m {𝐴}) → 𝑓 Fn {𝐴})
4442, 43syl 17 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
451sselda 3983 . . . . . . . . . . . 12 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵m {𝐴}))
46 elmapfn 8905 . . . . . . . . . . . 12 (𝑔 ∈ (𝐵m {𝐴}) → 𝑔 Fn {𝐴})
4745, 46syl 17 . . . . . . . . . . 11 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
48473adant3 1133 . . . . . . . . . 10 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
49483adant1r 1178 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5040, 41, 44, 49fsneqrn 45216 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓𝐴) ∈ ran 𝑔))
5138, 50mpbird 257 . . . . . . 7 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
52 simp2 1138 . . . . . . 7 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5351, 52eqeltrd 2841 . . . . . 6 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
5453rexlimdv3a 3159 . . . . 5 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
5537, 54mpd 15 . . . 4 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝑓𝐶)
5625, 55impbida 801 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
5756alrimiv 1927 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
58 nfcv 2905 . . 3 𝑓𝐶
59 ssmapsn.f . . . 4 𝑓𝐷
60 nfcv 2905 . . . 4 𝑓m
61 nfcv 2905 . . . 4 𝑓{𝐴}
6259, 60, 61nfov 7461 . . 3 𝑓(𝐷m {𝐴})
6358, 62cleqf 2934 . 2 (𝐶 = (𝐷m {𝐴}) ↔ ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
6457, 63sylibr 234 1 (𝜑𝐶 = (𝐷m {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087  wal 1538   = wceq 1540  wcel 2108  wnfc 2890  wral 3061  wrex 3070  Vcvv 3480  wss 3951  {csn 4626   ciun 4991  ran crn 5686   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868
This theorem is referenced by:  vonvolmbllem  46675  vonvolmbl2  46678  vonvol2  46679
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