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Theorem ssmapsn 40053
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 3761 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
3 elmapi 8082 . . . . . . . 8 (𝑓 ∈ (𝐵𝑚 {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 6224 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . . 8 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovexd 6876 . . . . . . . . . . 11 (𝜑 → (𝐵𝑚 {𝐴}) ∈ V)
98, 1ssexd 4966 . . . . . . . . . 10 (𝜑𝐶 ∈ V)
10 rnexg 7296 . . . . . . . . . . . 12 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 3069 . . . . . . . . . . 11 𝑓𝐶 ran 𝑓 ∈ V
1211a1i 11 . . . . . . . . . 10 (𝜑 → ∀𝑓𝐶 ran 𝑓 ∈ V)
139, 12jca 507 . . . . . . . . 9 (𝜑 → (𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V))
14 iunexg 7341 . . . . . . . . 9 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
1513, 14syl 17 . . . . . . . 8 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
167, 15eqeltrd 2844 . . . . . . 7 (𝜑𝐷 ∈ V)
1716adantr 472 . . . . . 6 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
18 ssiun2 4719 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1918adantl 473 . . . . . . . 8 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
20 ssmapsn.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
21 snidg 4364 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2322adantr 472 . . . . . . . . 9 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
24 fnfvelrn 6546 . . . . . . . . 9 ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓𝐴) ∈ ran 𝑓)
255, 23, 24syl2anc 579 . . . . . . . 8 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2619, 25sseldd 3762 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2726, 6syl6eleqr 2855 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
285, 17, 27elmapsnd 40041 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
2928ex 401 . . . 4 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
3016adantr 472 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐷 ∈ V)
31 snex 5064 . . . . . . . . . 10 {𝐴} ∈ V
3231a1i 11 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → {𝐴} ∈ V)
33 simpr 477 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
3422adantr 472 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐴 ∈ {𝐴})
3530, 32, 33, 34fvmap 40034 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝐷)
366idi 2 . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
37 rneq 5519 . . . . . . . . . 10 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3837cbviunv 4715 . . . . . . . . 9 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
3936, 38eqtri 2787 . . . . . . . 8 𝐷 = 𝑔𝐶 ran 𝑔
4035, 39syl6eleq 2854 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
41 eliun 4680 . . . . . . 7 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
4240, 41sylib 209 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
43 simp3 1168 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
44 simp1l 1254 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4544, 20syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
46 eqid 2765 . . . . . . . . . . 11 {𝐴} = {𝐴}
47 simp1r 1255 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
48 elmapfn 8083 . . . . . . . . . . . 12 (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓 Fn {𝐴})
4947, 48syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
501sselda 3761 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵𝑚 {𝐴}))
51 elmapfn 8083 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐵𝑚 {𝐴}) → 𝑔 Fn {𝐴})
5250, 51syl 17 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
53523adant3 1162 . . . . . . . . . . . 12 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
54533adant1r 1223 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5545, 46, 49, 54fsneqrn 40048 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓𝐴) ∈ ran 𝑔))
5643, 55mpbird 248 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
57 simp2 1167 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5856, 57eqeltrd 2844 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
59583exp 1148 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑔𝐶 → ((𝑓𝐴) ∈ ran 𝑔𝑓𝐶)))
6059rexlimdv 3177 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
6142, 60mpd 15 . . . . 5 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓𝐶)
6261ex 401 . . . 4 (𝜑 → (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓𝐶))
6329, 62impbid 203 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
6463alrimiv 2022 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
65 nfcv 2907 . . 3 𝑓𝐶
66 ssmapsn.f . . . 4 𝑓𝐷
67 nfcv 2907 . . . 4 𝑓𝑚
68 nfcv 2907 . . . 4 𝑓{𝐴}
6966, 67, 68nfov 6872 . . 3 𝑓(𝐷𝑚 {𝐴})
7065, 69dfcleqf 39906 . 2 (𝐶 = (𝐷𝑚 {𝐴}) ↔ ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
7164, 70sylibr 225 1 (𝜑𝐶 = (𝐷𝑚 {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107  wal 1650   = wceq 1652  wcel 2155  wnfc 2894  wral 3055  wrex 3056  Vcvv 3350  wss 3732  {csn 4334   ciun 4676  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062
This theorem is referenced by:  vonvolmbllem  41514  vonvolmbl2  41517  vonvol2  41518
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