Theorem ssmapsn 45571

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	⊢ Ⅎ𝑓𝐷
ssmapsn.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
ssmapsn.c	⊢ (𝜑 → 𝐶 ⊆ (𝐵 ↑m {𝐴}))
ssmapsn.d	⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓

Assertion

Ref	Expression
ssmapsn	⊢ (𝜑 → 𝐶 = (𝐷 ↑m {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.

Step	Hyp	Ref	Expression
1		ssmapsn.c	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ⊆ (𝐵 ↑m {𝐴}))
2	1	sselda 3935	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐵 ↑m {𝐴}))
3		elmapi 8798	. . . . . . 7 ⊢ (𝑓 ∈ (𝐵 ↑m {𝐴}) → 𝑓:{𝐴}⟶𝐵)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓:{𝐴}⟶𝐵)
5	4	ffnd 6671	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 Fn {𝐴})
6		ssmapsn.d	. . . . . . . 8 ⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓
7	6	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓)
8		ovexd 7403	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ∈ V)
9	8, 1	ssexd 5271	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
10		rnexg 7854	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ∈ V)
11	10	rgen 3054	. . . . . . . 8 ⊢ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V
12		iunexg 7917	. . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
13	9, 11, 12	sylancl 587	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
14	7, 13	eqeltrd 2837	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ V)
15	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐷 ∈ V)
16		ssiun2 5005	. . . . . . . 8 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
17	16	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
18		ssmapsn.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
19		snidg 4619	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
20	18, 19	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
21	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐴 ∈ {𝐴})
22	5, 21	fnfvelrnd 7036	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ran 𝑓)
23	17, 22	sseldd 3936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
24	23, 6	eleqtrrdi 2848	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ 𝐷)
25	5, 15, 24	elmapsnd 45559	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
26	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐷 ∈ V)
27		snex 5385	. . . . . . . . 9 ⊢ {𝐴} ∈ V
28	27	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → {𝐴} ∈ V)
29		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
30	20	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐴 ∈ {𝐴})
31	26, 28, 29, 30	fvmap 45553	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ 𝐷)
32		rneq 5893	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
33	32	cbviunv 4996	. . . . . . . 8 ⊢ ∪ 𝑓 ∈ 𝐶 ran 𝑓 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
34	6, 33	eqtri 2760	. . . . . . 7 ⊢ 𝐷 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
35	31, 34	eleqtrdi 2847	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔)
36		eliun 4952	. . . . . 6 ⊢ ((𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔 ↔ ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔)
37	35, 36	sylib 218	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔)
38		simp3 1139	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓‘𝐴) ∈ ran 𝑔)
39		simp1l 1199	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝜑)
40	39, 18	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝐴 ∈ 𝑉)
41		eqid 2737	. . . . . . . . 9 ⊢ {𝐴} = {𝐴}
42		simp1r 1200	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
43		elmapfn 8814	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝐷 ↑m {𝐴}) → 𝑓 Fn {𝐴})
44	42, 43	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
45	1	sselda 3935	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ (𝐵 ↑m {𝐴}))
46		elmapfn 8814	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝐵 ↑m {𝐴}) → 𝑔 Fn {𝐴})
47	45, 46	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 Fn {𝐴})
48	47	3adant3 1133	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
49	48	3adant1r 1179	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
50	40, 41, 44, 49	fsneqrn 45566	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓‘𝐴) ∈ ran 𝑔))
51	38, 50	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
52		simp2 1138	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 ∈ 𝐶)
53	51, 52	eqeltrd 2837	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ 𝐶)
54	53	rexlimdv3a 3143	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔 → 𝑓 ∈ 𝐶))
55	37, 54	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ 𝐶)
56	25, 55	impbida 801	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
57	56	alrimiv 1929	. 2 ⊢ (𝜑 → ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
58		nfcv 2899	. . 3 ⊢ Ⅎ𝑓𝐶
59		ssmapsn.f	. . . 4 ⊢ Ⅎ𝑓𝐷
60		nfcv 2899	. . . 4 ⊢ Ⅎ𝑓 ↑m
61		nfcv 2899	. . . 4 ⊢ Ⅎ𝑓{𝐴}
62	59, 60, 61	nfov 7398	. . 3 ⊢ Ⅎ𝑓(𝐷 ↑m {𝐴})
63	58, 62	cleqf 2928	. 2 ⊢ (𝐶 = (𝐷 ↑m {𝐴}) ↔ ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
64	57, 63	sylibr 234	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↑m {𝐴}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542   ∈ wcel 2114  Ⅎwnfc 2884  ∀wral 3052  ∃wrex 3062  Vcvv 3442   ⊆ wss 3903  {csn 4582  ∪ ciun 4948  ran crn 5633   Fn wfn 6495  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368   ↑_m cmap 8775

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-map 8777

This theorem is referenced by:  vonvolmbllem  47015  vonvolmbl2  47018  vonvol2  47019