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 41347
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 3971 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵m {𝐴}))
3 elmapi 8418 . . . . . . . 8 (𝑓 ∈ (𝐵m {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 6512 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . . 8 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovexd 7183 . . . . . . . . . . 11 (𝜑 → (𝐵m {𝐴}) ∈ V)
98, 1ssexd 5225 . . . . . . . . . 10 (𝜑𝐶 ∈ V)
10 rnexg 7602 . . . . . . . . . . . 12 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 3153 . . . . . . . . . . 11 𝑓𝐶 ran 𝑓 ∈ V
1211a1i 11 . . . . . . . . . 10 (𝜑 → ∀𝑓𝐶 ran 𝑓 ∈ V)
139, 12jca 512 . . . . . . . . 9 (𝜑 → (𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V))
14 iunexg 7655 . . . . . . . . 9 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
1513, 14syl 17 . . . . . . . 8 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
167, 15eqeltrd 2918 . . . . . . 7 (𝜑𝐷 ∈ V)
1716adantr 481 . . . . . 6 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
18 ssiun2 4968 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1918adantl 482 . . . . . . . 8 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
20 ssmapsn.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
21 snidg 4596 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2322adantr 481 . . . . . . . . 9 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
24 fnfvelrn 6844 . . . . . . . . 9 ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓𝐴) ∈ ran 𝑓)
255, 23, 24syl2anc 584 . . . . . . . 8 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2619, 25sseldd 3972 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2726, 6syl6eleqr 2929 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
285, 17, 27elmapsnd 41335 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷m {𝐴}))
2928ex 413 . . . 4 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
3016adantr 481 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐷 ∈ V)
31 snex 5328 . . . . . . . . . 10 {𝐴} ∈ V
3231a1i 11 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → {𝐴} ∈ V)
33 simpr 485 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝑓 ∈ (𝐷m {𝐴}))
3422adantr 481 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝐴 ∈ {𝐴})
3530, 32, 33, 34fvmap 41328 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝐷)
366idi 1 . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
37 rneq 5805 . . . . . . . . . 10 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3837cbviunv 4962 . . . . . . . . 9 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
3936, 38eqtri 2849 . . . . . . . 8 𝐷 = 𝑔𝐶 ran 𝑔
4035, 39syl6eleq 2928 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
41 eliun 4921 . . . . . . 7 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
4240, 41sylib 219 . . . . . 6 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
43 simp3 1132 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
44 simp1l 1191 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4544, 20syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
46 eqid 2826 . . . . . . . . . . 11 {𝐴} = {𝐴}
47 simp1r 1192 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷m {𝐴}))
48 elmapfn 8419 . . . . . . . . . . . 12 (𝑓 ∈ (𝐷m {𝐴}) → 𝑓 Fn {𝐴})
4947, 48syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
501sselda 3971 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵m {𝐴}))
51 elmapfn 8419 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐵m {𝐴}) → 𝑔 Fn {𝐴})
5250, 51syl 17 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
53523adant3 1126 . . . . . . . . . . . 12 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
54533adant1r 1171 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5545, 46, 49, 54fsneqrn 41342 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓𝐴) ∈ ran 𝑔))
5643, 55mpbird 258 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
57 simp2 1131 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5856, 57eqeltrd 2918 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷m {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
59583exp 1113 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (𝑔𝐶 → ((𝑓𝐴) ∈ ran 𝑔𝑓𝐶)))
6059rexlimdv 3288 . . . . . 6 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
6142, 60mpd 15 . . . . 5 ((𝜑𝑓 ∈ (𝐷m {𝐴})) → 𝑓𝐶)
6261ex 413 . . . 4 (𝜑 → (𝑓 ∈ (𝐷m {𝐴}) → 𝑓𝐶))
6329, 62impbid 213 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
6463alrimiv 1921 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
65 nfcv 2982 . . 3 𝑓𝐶
66 ssmapsn.f . . . 4 𝑓𝐷
67 nfcv 2982 . . . 4 𝑓m
68 nfcv 2982 . . . 4 𝑓{𝐴}
6966, 67, 68nfov 7178 . . 3 𝑓(𝐷m {𝐴})
7065, 69cleqf 3015 . 2 (𝐶 = (𝐷m {𝐴}) ↔ ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷m {𝐴})))
7164, 70sylibr 235 1 (𝜑𝐶 = (𝐷m {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081  wal 1528   = wceq 1530  wcel 2107  wnfc 2966  wral 3143  wrex 3144  Vcvv 3500  wss 3940  {csn 4564   ciun 4917  ran crn 5555   Fn wfn 6347  wf 6348  cfv 6352  (class class class)co 7148  m cmap 8396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-map 8398
This theorem is referenced by:  vonvolmbllem  42811  vonvolmbl2  42814  vonvol2  42815
  Copyright terms: Public domain W3C validator