Theorem unirnmapsn 39923

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 5057	. . . . 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 39917	. 2 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑𝑚 𝐶))
7		simpl 474	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝜑)
8		equid 2094	. . . . . . . . 9 ⊢ 𝑔 = 𝑔
9		rnuni 5702	. . . . . . . . . 10 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
10	9	oveq1i 6824	. . . . . . . . 9 ⊢ (ran ∪ 𝑋 ↑𝑚 𝐶) = (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)
11	8, 10	eleq12i 2832	. . . . . . . 8 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) ↔ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
12	11	biimpi 206	. . . . . . 7 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
13	12	adantl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
14		ovexd 6844	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ↑𝑚 𝐶) ∈ V)
15	14, 5	ssexd 4957	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ V)
16		rnexg 7264	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑋 → ran 𝑓 ∈ V)
17	16	rgen 3060	. . . . . . . . . . . 12 ⊢ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V
18	17	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
19		iunexg 7309	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
20	15, 18, 19	syl2anc 696	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
21	20, 4	elmapd 8039	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶) ↔ 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓))
22	21	biimpa 502	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓)
23		unirnmapsn.A	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
24		snidg 4351	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
25	23, 24	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
26	25, 1	syl6eleqr 2850	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
27	26	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝐴 ∈ 𝐶)
28	22, 27	ffvelrnd 6524	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → (𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
29		eliun 4676	. . . . . . 7 ⊢ ((𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
30	28, 29	sylib 208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
31	7, 13, 30	syl2anc 696	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
32		elmapfn 8048	. . . . . . . 8 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) → 𝑔 Fn 𝐶)
33	32	adantl 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 Fn 𝐶)
34		simp3 1133	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ ran 𝑓)
35	23	3ad2ant1 1128	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
36	1	oveq2i 6825	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ↑𝑚 𝐶) = (𝐵 ↑𝑚 {𝐴})
37	5, 36	syl6sseq 3792	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴}))
38	37	adantr 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴}))
39		simpr 479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
40	38, 39	sseldd 3745	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 {𝐴}))
41		unirnmapsn.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
42	41	adantr 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐵 ∈ 𝑊)
43	2	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝐴} ∈ V)
44	42, 43	elmapd 8039	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ∈ (𝐵 ↑𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
45	40, 44	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶𝐵)
46	45	3adant3 1127	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
47	35, 46	rnsnf 39887	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓‘𝐴)})
48	34, 47	eleqtrd 2841	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ {(𝑓‘𝐴)})
49		fvex 6363	. . . . . . . . . . . . 13 ⊢ (𝑔‘𝐴) ∈ V
50	49	elsn 4336	. . . . . . . . . . . 12 ⊢ ((𝑔‘𝐴) ∈ {(𝑓‘𝐴)} ↔ (𝑔‘𝐴) = (𝑓‘𝐴))
51	48, 50	sylib 208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
52	51	3adant1r 1188	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
53	23	adantr 472	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → 𝐴 ∈ 𝑉)
54	53	3ad2ant1 1128	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
55		simp1r 1241	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
56	40, 36	syl6eleqr 2850	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 𝐶))
57		elmapfn 8048	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐶) → 𝑓 Fn 𝐶)
58	56, 57	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
59	58	adantlr 753	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
60	59	3adant3 1127	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
61	54, 1, 55, 60	fsneq 39915	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔‘𝐴) = (𝑓‘𝐴)))
62	52, 61	mpbird 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
63		simp2 1132	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 ∈ 𝑋)
64	62, 63	eqeltrd 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 ∈ 𝑋)
65	64	3exp 1113	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
66	7, 33, 65	syl2anc 696	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
67	66	rexlimdv 3168	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → (∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋))
68	31, 67	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 ∈ 𝑋)
69	68	ralrimiva 3104	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)𝑔 ∈ 𝑋)
70		dfss3 3733	. . 3 ⊢ ((ran ∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)𝑔 ∈ 𝑋)
71	69, 70	sylibr 224	. 2 ⊢ (𝜑 → (ran ∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋)
72	6, 71	eqssd 3761	1 ⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑𝑚 𝐶))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  ∃wrex 3051  Vcvv 3340   ⊆ wss 3715  {csn 4321  ∪ cuni 4588  ∪ ciun 4672  ran crn 5267   Fn wfn 6044  ⟶wf 6045  ‘cfv 6049  (class class class)co 6814   ↑_𝑚 cmap 8025

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-map 8027

This theorem is referenced by: (None)