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Theorem ssmapsn 39919
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 3752 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
3 elmapi 8029 . . . . . . . 8 (𝑓 ∈ (𝐵𝑚 {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 6184 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . . 8 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovexd 6823 . . . . . . . . . . 11 (𝜑 → (𝐵𝑚 {𝐴}) ∈ V)
98, 1ssexd 4939 . . . . . . . . . 10 (𝜑𝐶 ∈ V)
10 rnexg 7243 . . . . . . . . . . . 12 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 3071 . . . . . . . . . . 11 𝑓𝐶 ran 𝑓 ∈ V
1211a1i 11 . . . . . . . . . 10 (𝜑 → ∀𝑓𝐶 ran 𝑓 ∈ V)
139, 12jca 501 . . . . . . . . 9 (𝜑 → (𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V))
14 iunexg 7288 . . . . . . . . 9 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
1513, 14syl 17 . . . . . . . 8 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
167, 15eqeltrd 2850 . . . . . . 7 (𝜑𝐷 ∈ V)
1716adantr 466 . . . . . 6 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
18 ssiun2 4697 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1918adantl 467 . . . . . . . 8 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
20 ssmapsn.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
21 snidg 4345 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2322adantr 466 . . . . . . . . 9 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
24 fnfvelrn 6497 . . . . . . . . 9 ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓𝐴) ∈ ran 𝑓)
255, 23, 24syl2anc 573 . . . . . . . 8 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2619, 25sseldd 3753 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2726, 6syl6eleqr 2861 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
285, 17, 27elmapsnd 39907 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
2928ex 397 . . . 4 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
3016adantr 466 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐷 ∈ V)
31 snex 5036 . . . . . . . . . 10 {𝐴} ∈ V
3231a1i 11 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → {𝐴} ∈ V)
33 simpr 471 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
3422adantr 466 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐴 ∈ {𝐴})
3530, 32, 33, 34fvmap 39900 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝐷)
366idi 2 . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
37 rneq 5487 . . . . . . . . . 10 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3837cbviunv 4693 . . . . . . . . 9 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
3936, 38eqtri 2793 . . . . . . . 8 𝐷 = 𝑔𝐶 ran 𝑔
4035, 39syl6eleq 2860 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
41 eliun 4658 . . . . . . 7 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
4240, 41sylib 208 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
43 simp3 1132 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
44 simp1l 1239 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4544, 20syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
46 eqid 2771 . . . . . . . . . . 11 {𝐴} = {𝐴}
47 simp1r 1240 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
48 elmapfn 8030 . . . . . . . . . . . 12 (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓 Fn {𝐴})
4947, 48syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
501sselda 3752 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵𝑚 {𝐴}))
51 elmapfn 8030 . . . . . . . . . . . . . 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 1187 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5545, 46, 49, 54fsneqrn 39914 . . . . . . . . . 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 1131 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5856, 57eqeltrd 2850 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
59583exp 1112 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑔𝐶 → ((𝑓𝐴) ∈ ran 𝑔𝑓𝐶)))
6059rexlimdv 3178 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
6142, 60mpd 15 . . . . 5 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓𝐶)
6261ex 397 . . . 4 (𝜑 → (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓𝐶))
6329, 62impbid 202 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
6463alrimiv 2007 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
65 nfcv 2913 . . 3 𝑓𝐶
66 ssmapsn.f . . . 4 𝑓𝐷
67 nfcv 2913 . . . 4 𝑓𝑚
68 nfcv 2913 . . . 4 𝑓{𝐴}
6966, 67, 68nfov 6819 . . 3 𝑓(𝐷𝑚 {𝐴})
7065, 69dfcleqf 39769 . 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 382  w3a 1071  wal 1629   = wceq 1631  wcel 2145  wnfc 2900  wral 3061  wrex 3062  Vcvv 3351  wss 3723  {csn 4316   ciun 4654  ran crn 5250   Fn wfn 6024  wf 6025  cfv 6029  (class class class)co 6791  𝑚 cmap 8007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-f1 6034  df-fo 6035  df-f1o 6036  df-fv 6037  df-ov 6794  df-oprab 6795  df-mpt2 6796  df-1st 7313  df-2nd 7314  df-map 8009
This theorem is referenced by:  vonvolmbllem  41387  vonvolmbl2  41390  vonvol2  41391
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