Theorem ssmapsn 41485

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 3970	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐵 ↑m {𝐴}))
3		elmapi 8431	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐵 ↑m {𝐴}) → 𝑓:{𝐴}⟶𝐵)
4	2, 3	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓:{𝐴}⟶𝐵)
5	4	ffnd 6518	. . . . . 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 7194	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 ↑m {𝐴}) ∈ V)
9	8, 1	ssexd 5231	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ V)
10		rnexg 7617	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ∈ V)
11	10	rgen 3151	. . . . . . . . . . 11 ⊢ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V
12	11	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V)
13	9, 12	jca 514	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ∈ V ∧ ∀𝑓 ∈ 𝐶 ran 𝑓 ∈ V))
14		iunexg 7667	. . . . . . . . 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 2916	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ V)
17	16	adantr 483	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐷 ∈ V)
18		ssiun2 4974	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
19	18	adantl 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → ran 𝑓 ⊆ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
20		ssmapsn.a	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
21		snidg 4602	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
22	20, 21	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
23	22	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝐴 ∈ {𝐴})
24		fnfvelrn 6851	. . . . . . . . 9 ⊢ ((𝑓 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → (𝑓‘𝐴) ∈ ran 𝑓)
25	5, 23, 24	syl2anc 586	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ran 𝑓)
26	19, 25	sseldd 3971	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ ∪ 𝑓 ∈ 𝐶 ran 𝑓)
27	26, 6	eleqtrrdi 2927	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → (𝑓‘𝐴) ∈ 𝐷)
28	5, 17, 27	elmapsnd 41473	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐶) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
29	28	ex 415	. . . 4 ⊢ (𝜑 → (𝑓 ∈ 𝐶 → 𝑓 ∈ (𝐷 ↑m {𝐴})))
30	16	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐷 ∈ V)
31		snex 5335	. . . . . . . . . 10 ⊢ {𝐴} ∈ V
32	31	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → {𝐴} ∈ V)
33		simpr 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
34	22	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → 𝐴 ∈ {𝐴})
35	30, 32, 33, 34	fvmap 41466	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ 𝐷)
36	6	idi 1	. . . . . . . . 9 ⊢ 𝐷 = ∪ 𝑓 ∈ 𝐶 ran 𝑓
37		rneq 5809	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
38	37	cbviunv 4968	. . . . . . . . 9 ⊢ ∪ 𝑓 ∈ 𝐶 ran 𝑓 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
39	36, 38	eqtri 2847	. . . . . . . 8 ⊢ 𝐷 = ∪ 𝑔 ∈ 𝐶 ran 𝑔
40	35, 39	eleqtrdi 2926	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔)
41		eliun 4926	. . . . . . 7 ⊢ ((𝑓‘𝐴) ∈ ∪ 𝑔 ∈ 𝐶 ran 𝑔 ↔ ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔)
42	40, 41	sylib 220	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → ∃𝑔 ∈ 𝐶 (𝑓‘𝐴) ∈ ran 𝑔)
43		simp3 1134	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓‘𝐴) ∈ ran 𝑔)
44		simp1l 1193	. . . . . . . . . . . 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 2824	. . . . . . . . . . 11 ⊢ {𝐴} = {𝐴}
47		simp1r 1194	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ (𝐷 ↑m {𝐴}))
48		elmapfn 8432	. . . . . . . . . . . 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 3970	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 ∈ (𝐵 ↑m {𝐴}))
51		elmapfn 8432	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ (𝐵 ↑m {𝐴}) → 𝑔 Fn {𝐴})
52	50, 51	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶) → 𝑔 Fn {𝐴})
53	52	3adant3 1128	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
54	53	3adant1r 1173	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 Fn {𝐴})
55	45, 46, 49, 54	fsneqrn 41480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → (𝑓 = 𝑔 ↔ (𝑓‘𝐴) ∈ ran 𝑔))
56	43, 55	mpbird 259	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 = 𝑔)
57		simp2 1133	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑔 ∈ 𝐶)
58	56, 57	eqeltrd 2916	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) ∧ 𝑔 ∈ 𝐶 ∧ (𝑓‘𝐴) ∈ ran 𝑔) → 𝑓 ∈ 𝐶)
59	58	3exp 1115	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝐷 ↑m {𝐴})) → (𝑔 ∈ 𝐶 → ((𝑓‘𝐴) ∈ ran 𝑔 → 𝑓 ∈ 𝐶)))
60	59	rexlimdv 3286	. . . . . 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 415	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐷 ↑m {𝐴}) → 𝑓 ∈ 𝐶))
63	29, 62	impbid 214	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
64	63	alrimiv 1927	. 2 ⊢ (𝜑 → ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
65		nfcv 2980	. . 3 ⊢ Ⅎ𝑓𝐶
66		ssmapsn.f	. . . 4 ⊢ Ⅎ𝑓𝐷
67		nfcv 2980	. . . 4 ⊢ Ⅎ𝑓 ↑m
68		nfcv 2980	. . . 4 ⊢ Ⅎ𝑓{𝐴}
69	66, 67, 68	nfov 7189	. . 3 ⊢ Ⅎ𝑓(𝐷 ↑m {𝐴})
70	65, 69	cleqf 3013	. 2 ⊢ (𝐶 = (𝐷 ↑m {𝐴}) ↔ ∀𝑓(𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐷 ↑m {𝐴})))
71	64, 70	sylibr 236	1 ⊢ (𝜑 → 𝐶 = (𝐷 ↑m {𝐴}))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1534   = wceq 1536   ∈ wcel 2113  Ⅎwnfc 2964  ∀wral 3141  ∃wrex 3142  Vcvv 3497   ⊆ wss 3939  {csn 4570  ∪ ciun 4922  ran crn 5559   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159   ↑_m cmap 8409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-map 8411

This theorem is referenced by:  vonvolmbllem  42949  vonvolmbl2  42952  vonvol2  42953