Theorem unirnmapsn 45659

Description: Equality theorem for a subset of a set exponentiation, where the exponent is a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
unirnmapsn.A	⊢ (𝜑 → 𝐴 ∈ 𝑉)
unirnmapsn.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
unirnmapsn.C	⊢ 𝐶 = {𝐴}
unirnmapsn.x	⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m 𝐶))

Assertion

Ref	Expression
unirnmapsn	⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑m 𝐶))

Proof of Theorem unirnmapsn

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unirnmapsn.C	. . . . 5 ⊢ 𝐶 = {𝐴}
2		snex 5368	. . . . 5 ⊢ {𝐴} ∈ V
3	1, 2	eqeltri 2835	. . . 4 ⊢ 𝐶 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		unirnmapsn.x	. . 3 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m 𝐶))
6	4, 5	unirnmap 45653	. 2 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐶))
7		simpl 483	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝜑)
8		equid 2019	. . . . . 6 ⊢ 𝑔 = 𝑔
9		rnuni 6099	. . . . . . 7 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
10	9	oveq1i 7366	. . . . . 6 ⊢ (ran ∪ 𝑋 ↑m 𝐶) = (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)
11	8, 10	eleq12i 2832	. . . . 5 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶) ↔ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶))
12	11	bilani 505	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶))
13		ovexd 7391	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ↑m 𝐶) ∈ V)
14	13, 5	ssexd 5252	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V)
15		rnexg 7842	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 → ran 𝑓 ∈ V)
16	15	rgen 3055	. . . . . . . . . 10 ⊢ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
18		iunexg 7905	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
19	14, 17, 18	syl2anc 590	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
20	19, 4	elmapd 8777	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶) ↔ 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓))
21	20	biimpa 477	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓)
22		unirnmapsn.A	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
23		snidg 4592	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
24	22, 23	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
25	24, 1	eleqtrrdi 2850	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
26	25	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → 𝐴 ∈ 𝐶)
27	21, 26	ffvelcdmd 7026	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → (𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
28		eliun 4925	. . . . 5 ⊢ ((𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
29	27, 28	sylib 219	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
30	7, 12, 29	syl2anc 590	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
31		elmapfn 8802	. . . . . 6 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶) → 𝑔 Fn 𝐶)
32	31	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 Fn 𝐶)
33		simp3 1144	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ ran 𝑓)
34	22	3ad2ant1 1139	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
35	1	oveq2i 7367	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ↑m 𝐶) = (𝐵 ↑m {𝐴})
36	5, 35	sseqtrdi 3955	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m {𝐴}))
37	36	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (𝐵 ↑m {𝐴}))
38		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
39	37, 38	sseldd 3916	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑m {𝐴}))
40		unirnmapsn.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
41	40	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐵 ∈ 𝑊)
42	2	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝐴} ∈ V)
43	41, 42	elmapd 8777	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ∈ (𝐵 ↑m {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
44	39, 43	mpbid 233	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶𝐵)
45	44	3adant3 1138	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
46	34, 45	rnsnf 45631	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓‘𝐴)})
47	33, 46	eleqtrd 2841	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ {(𝑓‘𝐴)})
48		fvex 6840	. . . . . . . . . . 11 ⊢ (𝑔‘𝐴) ∈ V
49	48	elsn 4570	. . . . . . . . . 10 ⊢ ((𝑔‘𝐴) ∈ {(𝑓‘𝐴)} ↔ (𝑔‘𝐴) = (𝑓‘𝐴))
50	47, 49	sylib 219	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
51	50	3adant1r 1184	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
52	22	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → 𝐴 ∈ 𝑉)
53	52	3ad2ant1 1139	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
54		simp1r 1205	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
55	39, 35	eleqtrrdi 2850	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑m 𝐶))
56		elmapfn 8802	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐶) → 𝑓 Fn 𝐶)
57	55, 56	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
58	57	adantlr 721	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
59	58	3adant3 1138	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
60	53, 1, 54, 59	fsneq 6976	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔‘𝐴) = (𝑓‘𝐴)))
61	51, 60	mpbird 258	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
62		simp2 1143	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 ∈ 𝑋)
63	61, 62	eqeltrd 2839	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 ∈ 𝑋)
64	63	3exp 1125	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
65	7, 32, 64	syl2anc 590	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
66	65	rexlimdv 3138	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → (∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋))
67	30, 66	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 ∈ 𝑋)
68	6, 67	eqelssd 3936	1 ⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑m 𝐶))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  {csn 4555  ∪ cuni 4838  ∪ ciun 4921  ran crn 5619   Fn wfn 6480  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356   ↑_m cmap 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-map 8765

This theorem is referenced by: (None)