Theorem ssmapsn 45460

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 3933	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐵 ↑m {𝐴}))
3		elmapi 8786	. . . . . . 7 ⊢ (𝑓 ∈ (𝐵 ↑m {𝐴}) → 𝑓:{𝐴}⟶𝐵)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓:{𝐴}⟶𝐵)
5	4	ffnd 6663	. . . . 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 7393	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ∈ V)
9	8, 1	ssexd 5269	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ V)
10		rnexg 7844	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ∈ V)
11	10	rgen 3053	. . . . . . . 8 ⊢ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V
12		iunexg 7907	. . . . . . . 8 ⊢ ((𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
13	9, 11, 12	sylancl 586	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
14	7, 13	eqeltrd 2836	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ V)
15	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐷 ∈ V)
16		ssiun2 5003	. . . . . . . 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 4617	. . . . . . . . . 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 7027	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ran 𝑓)
23	17, 22	sseldd 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
24	23, 6	eleqtrrdi 2847	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ 𝐷)
25	5, 15, 24	elmapsnd 45448	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
26	14	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐷 ∈ V)
27		snex 5381	. . . . . . . . 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 45442	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ 𝐷)
32		rneq 5885	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
33	32	cbviunv 4994	. . . . . . . 8 ⊢ ∪ 𝑓 ∈ 𝐶 ran 𝑓 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
34	6, 33	eqtri 2759	. . . . . . 7 ⊢ 𝐷 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
35	31, 34	eleqtrdi 2846	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔)
36		eliun 4950	. . . . . 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 1138	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓‘𝐴) ∈ ran 𝑔)
39		simp1l 1198	. . . . . . . . . 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 2736	. . . . . . . . 9 ⊢ {𝐴} = {𝐴}
42		simp1r 1199	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
43		elmapfn 8802	. . . . . . . . . 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 3933	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ (𝐵 ↑m {𝐴}))
46		elmapfn 8802	. . . . . . . . . . . 12 ⊢ (𝑔 ∈ (𝐵 ↑m {𝐴}) → 𝑔 Fn {𝐴})
47	45, 46	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 Fn {𝐴})
48	47	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
49	48	3adant1r 1178	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
50	40, 41, 44, 49	fsneqrn 45455	. . . . . . . 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 1137	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 ∈ 𝐶)
53	51, 52	eqeltrd 2836	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ 𝐶)
54	53	rexlimdv3a 3141	. . . . 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 800	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
57	56	alrimiv 1928	. 2 ⊢ (𝜑 → ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
58		nfcv 2898	. . 3 ⊢ Ⅎ𝑓𝐶
59		ssmapsn.f	. . . 4 ⊢ Ⅎ𝑓𝐷
60		nfcv 2898	. . . 4 ⊢ Ⅎ𝑓 ↑m
61		nfcv 2898	. . . 4 ⊢ Ⅎ𝑓{𝐴}
62	59, 60, 61	nfov 7388	. . 3 ⊢ Ⅎ𝑓(𝐷 ↑m {𝐴})
63	58, 62	cleqf 2927	. 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 1086  ∀wal 1539   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2883  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ⊆ wss 3901  {csn 4580  ∪ ciun 4946  ran crn 5625   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358   ↑_m cmap 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-map 8765

This theorem is referenced by:  vonvolmbllem  46904  vonvolmbl2  46907  vonvol2  46908