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Theorem ssmapsn 45158
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 3994 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵m {𝐴}))
3 elmapi 8887 . . . . . . 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 7465 . . . . . . . . 9 (𝜑 → (𝐵m {𝐴}) ∈ V)
98, 1ssexd 5329 . . . . . . . 8 (𝜑𝐶 ∈ V)
10 rnexg 7924 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 3060 . . . . . . . 8 𝑓𝐶 ran 𝑓 ∈ V
12 iunexg 7986 . . . . . . . 8 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
139, 11, 12sylancl 586 . . . . . . 7 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
147, 13eqeltrd 2838 . . . . . 6 (𝜑𝐷 ∈ V)
1514adantr 480 . . . . 5 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
16 ssiun2 5051 . . . . . . . 8 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1716adantl 481 . . . . . . 7 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
18 ssmapsn.a . . . . . . . . . 10 (𝜑𝐴𝑉)
19 snidg 4664 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {𝐴})
2018, 19syl 17 . . . . . . . . 9 (𝜑𝐴 ∈ {𝐴})
2120adantr 480 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
225, 21fnfvelrnd 7101 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2317, 22sseldd 3995 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2423, 6eleqtrrdi 2849 . . . . 5 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
255, 15, 24elmapsnd 45146 . . . 4 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷m {𝐴}))
2614adantr 480 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐷 ∈ V)
27 snex 5441 . . . . . . . . 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 45140 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝐷)
32 rneq 5949 . . . . . . . . 9 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3332cbviunv 5044 . . . . . . . 8 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
346, 33eqtri 2762 . . . . . . 7 𝐷 = 𝑔𝐶 ran 𝑔
3531, 34eleqtrdi 2848 . . . . . 6 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
36 eliun 4999 . . . . . 6 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
3735, 36sylib 218 . . . . 5 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
38 simp3 1137 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
39 simp1l 1196 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4039, 18syl 17 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
41 eqid 2734 . . . . . . . . 9 {𝐴} = {𝐴}
42 simp1r 1197 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷m {𝐴}))
43 elmapfn 8903 . . . . . . . . . 10 (𝑓 ∈ (𝐷m {𝐴}) → 𝑓 Fn {𝐴})
4442, 43syl 17 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
451sselda 3994 . . . . . . . . . . . 12 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵m {𝐴}))
46 elmapfn 8903 . . . . . . . . . . . 12 (𝑔 ∈ (𝐵m {𝐴}) → 𝑔 Fn {𝐴})
4745, 46syl 17 . . . . . . . . . . 11 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
48473adant3 1131 . . . . . . . . . 10 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
49483adant1r 1176 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5040, 41, 44, 49fsneqrn 45153 . . . . . . . 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 1136 . . . . . . 7 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5351, 52eqeltrd 2838 . . . . . 6 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
5453rexlimdv3a 3156 . . . . 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 1924 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
58 nfcv 2902 . . 3 𝑓𝐶
59 ssmapsn.f . . . 4 𝑓𝐷
60 nfcv 2902 . . . 4 𝑓m
61 nfcv 2902 . . . 4 𝑓{𝐴}
6259, 60, 61nfov 7460 . . 3 𝑓(𝐷m {𝐴})
6358, 62cleqf 2931 . 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 1086  wal 1534   = wceq 1536  wcel 2105  wnfc 2887  wral 3058  wrex 3067  Vcvv 3477  wss 3962  {csn 4630   ciun 4995  ran crn 5689   Fn wfn 6557  wf 6558  cfv 6562  (class class class)co 7430  m cmap 8864
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-map 8866
This theorem is referenced by:  vonvolmbllem  46615  vonvolmbl2  46618  vonvol2  46619
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