Theorem ssmapsn 42645

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	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ⊆ (𝐵 ↑m {𝐴}))
2	1	sselda 3917	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐵 ↑m {𝐴}))
3		elmapi 8595	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐵 ↑m {𝐴}) → 𝑓:{𝐴}⟶𝐵)
4	2, 3	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓:{𝐴}⟶𝐵)
5	4	ffnd 6585	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 Fn {𝐴})
6		ssmapsn.d	. . . . . . . . 9 ⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓
7	6	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓)
8		ovexd 7290	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ∈ V)
9	8, 1	ssexd 5243	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ V)
10		rnexg 7725	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ∈ V)
11	10	rgen 3073	. . . . . . . . . . 11 ⊢ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V
12	11	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
13	9, 12	jca 511	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V))
14		iunexg 7779	. . . . . . . . 9 ⊢ ((𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
15	13, 14	syl 17	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
16	7, 15	eqeltrd 2839	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ V)
17	16	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐷 ∈ V)
18		ssiun2 4973	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
19	18	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
20		ssmapsn.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
21		snidg 4592	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐴 ∈ {𝐴})
24		fnfvelrn 6940	. . . . . . . . 9 ⊢ ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓‘𝐴) ∈ ran 𝑓)
25	5, 23, 24	syl2anc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ran 𝑓)
26	19, 25	sseldd 3918	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
27	26, 6	eleqtrrdi 2850	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ 𝐷)
28	5, 17, 27	elmapsnd 42633	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
29	28	ex 412	. . . 4 ⊢ (𝜑 → (𝑓 ∈ 𝐶 → 𝑓 ∈ (𝐷 ↑m {𝐴})))
30	16	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐷 ∈ V)
31		snex 5349	. . . . . . . . . 10 ⊢ {𝐴} ∈ V
32	31	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → {𝐴} ∈ V)
33		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
34	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐴 ∈ {𝐴})
35	30, 32, 33, 34	fvmap 42626	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ 𝐷)
36	6	idi 1	. . . . . . . . 9 ⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓
37		rneq 5834	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
38	37	cbviunv 4966	. . . . . . . . 9 ⊢ ∪ 𝑓 ∈ 𝐶 ran 𝑓 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
39	36, 38	eqtri 2766	. . . . . . . 8 ⊢ 𝐷 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
40	35, 39	eleqtrdi 2849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔)
41		eliun 4925	. . . . . . 7 ⊢ ((𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔 ↔ ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔)
42	40, 41	sylib 217	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔)
43		simp3 1136	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓‘𝐴) ∈ ran 𝑔)
44		simp1l 1195	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝜑)
45	44, 20	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝐴 ∈ 𝑉)
46		eqid 2738	. . . . . . . . . . 11 ⊢ {𝐴} = {𝐴}
47		simp1r 1196	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
48		elmapfn 8611	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐷 ↑m {𝐴}) → 𝑓 Fn {𝐴})
49	47, 48	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 Fn {𝐴})
50	1	sselda 3917	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ (𝐵 ↑m {𝐴}))
51		elmapfn 8611	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝐵 ↑m {𝐴}) → 𝑔 Fn {𝐴})
52	50, 51	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 Fn {𝐴})
53	52	3adant3 1130	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
54	53	3adant1r 1175	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
55	45, 46, 49, 54	fsneqrn 42640	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓‘𝐴) ∈ ran 𝑔))
56	43, 55	mpbird 256	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
57		simp2 1135	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 ∈ 𝐶)
58	56, 57	eqeltrd 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ 𝐶)
59	58	3exp 1117	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑔 ∈ 𝐶 → ((𝑓‘𝐴) ∈ ran 𝑔 → 𝑓 ∈ 𝐶)))
60	59	rexlimdv 3211	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔 → 𝑓 ∈ 𝐶))
61	42, 60	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ 𝐶)
62	61	ex 412	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐷 ↑m {𝐴}) → 𝑓 ∈ 𝐶))
63	29, 62	impbid 211	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
64	63	alrimiv 1931	. 2 ⊢ (𝜑 → ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
65		nfcv 2906	. . 3 ⊢ Ⅎ𝑓𝐶
66		ssmapsn.f	. . . 4 ⊢ Ⅎ𝑓𝐷
67		nfcv 2906	. . . 4 ⊢ Ⅎ𝑓 ↑m
68		nfcv 2906	. . . 4 ⊢ Ⅎ𝑓{𝐴}
69	66, 67, 68	nfov 7285	. . 3 ⊢ Ⅎ𝑓(𝐷 ↑m {𝐴})
70	65, 69	cleqf 2937	. 2 ⊢ (𝐶 = (𝐷 ↑m {𝐴}) ↔ ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
71	64, 70	sylibr 233	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↑m {𝐴}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539   ∈ wcel 2108  Ⅎwnfc 2886  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  {csn 4558  ∪ ciun 4921  ran crn 5581   Fn wfn 6413  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575

This theorem is referenced by:  vonvolmbllem  44088  vonvolmbl2  44091  vonvol2  44092