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Theorem ssmapsn 38913
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 3587 . . . . . . . 8 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐵𝑚 {𝐴}))
3 elmapi 7831 . . . . . . . 8 (𝑓 ∈ (𝐵𝑚 {𝐴}) → 𝑓:{𝐴}⟶𝐵)
42, 3syl 17 . . . . . . 7 ((𝜑𝑓𝐶) → 𝑓:{𝐴}⟶𝐵)
54ffnd 6008 . . . . . 6 ((𝜑𝑓𝐶) → 𝑓 Fn {𝐴})
6 ssmapsn.d . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
76a1i 11 . . . . . . . 8 (𝜑𝐷 = 𝑓𝐶 ran 𝑓)
8 ovexd 6640 . . . . . . . . . . 11 (𝜑 → (𝐵𝑚 {𝐴}) ∈ V)
98, 1ssexd 4770 . . . . . . . . . 10 (𝜑𝐶 ∈ V)
10 rnexg 7052 . . . . . . . . . . . 12 (𝑓𝐶 → ran 𝑓 ∈ V)
1110rgen 2917 . . . . . . . . . . 11 𝑓𝐶 ran 𝑓 ∈ V
1211a1i 11 . . . . . . . . . 10 (𝜑 → ∀𝑓𝐶 ran 𝑓 ∈ V)
139, 12jca 554 . . . . . . . . 9 (𝜑 → (𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V))
14 iunexg 7096 . . . . . . . . 9 ((𝐶 ∈ V ∧ ∀𝑓𝐶 ran 𝑓 ∈ V) → 𝑓𝐶 ran 𝑓 ∈ V)
1513, 14syl 17 . . . . . . . 8 (𝜑 𝑓𝐶 ran 𝑓 ∈ V)
167, 15eqeltrd 2698 . . . . . . 7 (𝜑𝐷 ∈ V)
1716adantr 481 . . . . . 6 ((𝜑𝑓𝐶) → 𝐷 ∈ V)
18 ssiun2 4534 . . . . . . . . 9 (𝑓𝐶 → ran 𝑓 𝑓𝐶 ran 𝑓)
1918adantl 482 . . . . . . . 8 ((𝜑𝑓𝐶) → ran 𝑓 𝑓𝐶 ran 𝑓)
20 ssmapsn.a . . . . . . . . . . 11 (𝜑𝐴𝑉)
21 snidg 4182 . . . . . . . . . . 11 (𝐴𝑉𝐴 ∈ {𝐴})
2220, 21syl 17 . . . . . . . . . 10 (𝜑𝐴 ∈ {𝐴})
2322adantr 481 . . . . . . . . 9 ((𝜑𝑓𝐶) → 𝐴 ∈ {𝐴})
24 fnfvelrn 6317 . . . . . . . . 9 ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓𝐴) ∈ ran 𝑓)
255, 23, 24syl2anc 692 . . . . . . . 8 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ ran 𝑓)
2619, 25sseldd 3588 . . . . . . 7 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝑓𝐶 ran 𝑓)
2726, 6syl6eleqr 2709 . . . . . 6 ((𝜑𝑓𝐶) → (𝑓𝐴) ∈ 𝐷)
285, 17, 27elmapsnd 38901 . . . . 5 ((𝜑𝑓𝐶) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
2928ex 450 . . . 4 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
3016adantr 481 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐷 ∈ V)
31 snex 4874 . . . . . . . . . 10 {𝐴} ∈ V
3231a1i 11 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → {𝐴} ∈ V)
33 simpr 477 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
3422adantr 481 . . . . . . . . 9 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝐴 ∈ {𝐴})
3530, 32, 33, 34fvmap 38892 . . . . . . . 8 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝐷)
366idi 2 . . . . . . . . 9 𝐷 = 𝑓𝐶 ran 𝑓
37 rneq 5316 . . . . . . . . . 10 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
3837cbviunv 4530 . . . . . . . . 9 𝑓𝐶 ran 𝑓 = 𝑔𝐶 ran 𝑔
3936, 38eqtri 2643 . . . . . . . 8 𝐷 = 𝑔𝐶 ran 𝑔
4035, 39syl6eleq 2708 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔)
41 eliun 4495 . . . . . . 7 ((𝑓𝐴) ∈ 𝑔𝐶 ran 𝑔 ↔ ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
4240, 41sylib 208 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → ∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔)
43 simp3 1061 . . . . . . . . . 10 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → (𝑓𝐴) ∈ ran 𝑔)
44 simp1l 1083 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝜑)
4544, 20syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝐴𝑉)
46 eqid 2621 . . . . . . . . . . 11 {𝐴} = {𝐴}
47 simp1r 1084 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷𝑚 {𝐴}))
48 elmapfn 7832 . . . . . . . . . . . 12 (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓 Fn {𝐴})
4947, 48syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
501sselda 3587 . . . . . . . . . . . . . 14 ((𝜑𝑔𝐶) → 𝑔 ∈ (𝐵𝑚 {𝐴}))
51 elmapfn 7832 . . . . . . . . . . . . . 14 (𝑔 ∈ (𝐵𝑚 {𝐴}) → 𝑔 Fn {𝐴})
5250, 51syl 17 . . . . . . . . . . . . 13 ((𝜑𝑔𝐶) → 𝑔 Fn {𝐴})
53523adant3 1079 . . . . . . . . . . . 12 ((𝜑𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
54533adant1r 1316 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
5545, 46, 49, 54fsneqrn 38908 . . . . . . . . . 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 1060 . . . . . . . . 9 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑔𝐶)
5856, 57eqeltrd 2698 . . . . . . . 8 (((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) ∧ 𝑔𝐶 ∧ (𝑓𝐴) ∈ ran 𝑔) → 𝑓𝐶)
59583exp 1261 . . . . . . 7 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (𝑔𝐶 → ((𝑓𝐴) ∈ ran 𝑔𝑓𝐶)))
6059rexlimdv 3024 . . . . . 6 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → (∃𝑔𝐶 (𝑓𝐴) ∈ ran 𝑔𝑓𝐶))
6142, 60mpd 15 . . . . 5 ((𝜑𝑓 ∈ (𝐷𝑚 {𝐴})) → 𝑓𝐶)
6261ex 450 . . . 4 (𝜑 → (𝑓 ∈ (𝐷𝑚 {𝐴}) → 𝑓𝐶))
6329, 62impbid 202 . . 3 (𝜑 → (𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
6463alrimiv 1852 . 2 (𝜑 → ∀𝑓(𝑓𝐶𝑓 ∈ (𝐷𝑚 {𝐴})))
65 nfcv 2761 . . 3 𝑓𝐶
66 ssmapsn.f . . . 4 𝑓𝐷
67 nfcv 2761 . . . 4 𝑓𝑚
68 nfcv 2761 . . . 4 𝑓{𝐴}
6966, 67, 68nfov 6636 . . 3 𝑓(𝐷𝑚 {𝐴})
7065, 69dfcleqf 38773 . 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 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  wnfc 2748  wral 2907  wrex 2908  Vcvv 3189  wss 3559  {csn 4153   ciun 4490  ran crn 5080   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  𝑚 cmap 7809
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-map 7811
This theorem is referenced by:  vonvolmbllem  40207  vonvolmbl2  40210  vonvol2  40211
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