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Theorem ssmapsn 39722
 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 3636 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
3 elmapi 7921 . . . . . . . 8 (𝑓 ∈ (𝐵𝑚 {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 6084 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . . 8 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovexd 6720 . . . . . . . . . . 11 (𝜑 → (𝐵𝑚 {𝐴}) ∈ V)
98, 1ssexd 4838 . . . . . . . . . 10 (𝜑𝐶 ∈ V)
10 rnexg 7140 . . . . . . . . . . . 12 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 2951 . . . . . . . . . . 11 𝑓𝐶 ran 𝑓 ∈ V
1211a1i 11 . . . . . . . . . 10 (𝜑 → ∀𝑓𝐶 ran 𝑓 ∈ V)
139, 12jca 553 . . . . . . . . 9 (𝜑 → (𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V))
14 iunexg 7185 . . . . . . . . 9 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
1513, 14syl 17 . . . . . . . 8 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
167, 15eqeltrd 2730 . . . . . . 7 (𝜑𝐷 ∈ V)
1716adantr 480 . . . . . 6 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
18 ssiun2 4595 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1918adantl 481 . . . . . . . 8 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
20 ssmapsn.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
21 snidg 4239 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2322adantr 480 . . . . . . . . 9 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
24 fnfvelrn 6396 . . . . . . . . 9 ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓𝐴) ∈ ran 𝑓)
255, 23, 24syl2anc 694 . . . . . . . 8 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2619, 25sseldd 3637 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2726, 6syl6eleqr 2741 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
285, 17, 27elmapsnd 39710 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
2928ex 449 . . . 4 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
3016adantr 480 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐷 ∈ V)
31 snex 4938 . . . . . . . . . 10 {𝐴} ∈ V
3231a1i 11 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → {𝐴} ∈ V)
33 simpr 476 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
3422adantr 480 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐴 ∈ {𝐴})
3530, 32, 33, 34fvmap 39701 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝐷)
366idi 2 . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
37 rneq 5383 . . . . . . . . . 10 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3837cbviunv 4591 . . . . . . . . 9 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
3936, 38eqtri 2673 . . . . . . . 8 𝐷 = 𝑔𝐶 ran 𝑔
4035, 39syl6eleq 2740 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
41 eliun 4556 . . . . . . 7 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
4240, 41sylib 208 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
43 simp3 1083 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
44 simp1l 1105 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4544, 20syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
46 eqid 2651 . . . . . . . . . . 11 {𝐴} = {𝐴}
47 simp1r 1106 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
48 elmapfn 7922 . . . . . . . . . . . 12 (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓 Fn {𝐴})
4947, 48syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
501sselda 3636 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵𝑚 {𝐴}))
51 elmapfn 7922 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐵𝑚 {𝐴}) → 𝑔 Fn {𝐴})
5250, 51syl 17 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
53523adant3 1101 . . . . . . . . . . . 12 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
54533adant1r 1359 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5545, 46, 49, 54fsneqrn 39717 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓𝐴) ∈ ran 𝑔))
5643, 55mpbird 247 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
57 simp2 1082 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5856, 57eqeltrd 2730 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
59583exp 1283 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑔𝐶 → ((𝑓𝐴) ∈ ran 𝑔𝑓𝐶)))
6059rexlimdv 3059 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
6142, 60mpd 15 . . . . 5 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓𝐶)
6261ex 449 . . . 4 (𝜑 → (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓𝐶))
6329, 62impbid 202 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
6463alrimiv 1895 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
65 nfcv 2793 . . 3 𝑓𝐶
66 ssmapsn.f . . . 4 𝑓𝐷
67 nfcv 2793 . . . 4 𝑓𝑚
68 nfcv 2793 . . . 4 𝑓{𝐴}
6966, 67, 68nfov 6716 . . 3 𝑓(𝐷𝑚 {𝐴})
7065, 69dfcleqf 39569 . 2 (𝐶 = (𝐷𝑚 {𝐴}) ↔ ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
7164, 70sylibr 224 1 (𝜑𝐶 = (𝐷𝑚 {𝐴}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054  ∀wal 1521   = wceq 1523   ∈ wcel 2030  Ⅎwnfc 2780  ∀wral 2941  ∃wrex 2942  Vcvv 3231   ⊆ wss 3607  {csn 4210  ∪ ciun 4552  ran crn 5144   Fn wfn 5921  ⟶wf 5922  ‘cfv 5926  (class class class)co 6690   ↑𝑚 cmap 7899 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901 This theorem is referenced by:  vonvolmbllem  41195  vonvolmbl2  41198  vonvol2  41199
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