Theorem unirnmapsn 46500

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 5404	. . . . 5 ⊢ {𝐴} ∈ V
3	1, 2	eqeltri 2860	. . . 4 ⊢ 𝐶 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		unirnmapsn.x	. . 3 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m 𝐶))
6	4, 5	unirnmap 46494	. 2 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐶))
7		simpl 488	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝜑)
8		equid 2047	. . . . . 6 ⊢ 𝑔 = 𝑔
9		rnuni 6158	. . . . . . 7 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
10	9	oveq1i 7461	. . . . . 6 ⊢ (ran ∪ 𝑋 ↑m 𝐶) = (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)
11	8, 10	eleq12i 2857	. . . . 5 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶) ↔ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶))
12	11	bilani 510	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶))
13		ovexd 7486	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ↑m 𝐶) ∈ V)
14	13, 5	ssexd 5293	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V)
15		rnexg 7948	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 → ran 𝑓 ∈ V)
16	15	rgen 3082	. . . . . . . . . 10 ⊢ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
18		iunexg 8009	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
19	14, 17, 18	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
20	19, 4	elmapd 8898	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶) ↔ 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓))
21	20	biimpa 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓)
22		unirnmapsn.A	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
23		snidg 4622	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
24	22, 23	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
25	24, 1	eleqtrrdi 2875	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
26	25	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → 𝐴 ∈ 𝐶)
27	21, 26	ffvelcdmd 7117	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → (𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
28		eliun 4960	. . . . 5 ⊢ ((𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
29	27, 28	sylib 221	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
30	7, 12, 29	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
31		elmapfn 8925	. . . . . 6 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶) → 𝑔 Fn 𝐶)
32	31	adantl 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 Fn 𝐶)
33		simp3 1156	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ ran 𝑓)
34	22	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
35	1	oveq2i 7462	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ↑m 𝐶) = (𝐵 ↑m {𝐴})
36	5, 35	sseqtrdi 3953	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m {𝐴}))
37	36	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (𝐵 ↑m {𝐴}))
38		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
39	37, 38	sseldd 3914	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑m {𝐴}))
40		unirnmapsn.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
41	40	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐵 ∈ 𝑊)
42	2	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝐴} ∈ V)
43	41, 42	elmapd 8898	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ∈ (𝐵 ↑m {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
44	39, 43	mpbid 235	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶𝐵)
45	44	3adant3 1150	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
46	34, 45	rnsnf 46472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓‘𝐴)})
47	33, 46	eleqtrd 2866	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ {(𝑓‘𝐴)})
48		fvex 6929	. . . . . . . . . . 11 ⊢ (𝑔‘𝐴) ∈ V
49	48	elsn 4599	. . . . . . . . . 10 ⊢ ((𝑔‘𝐴) ∈ {(𝑓‘𝐴)} ↔ (𝑔‘𝐴) = (𝑓‘𝐴))
50	47, 49	sylib 221	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
51	50	3adant1r 1196	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
52	22	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → 𝐴 ∈ 𝑉)
53	52	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
54		simp1r 1217	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
55	39, 35	eleqtrrdi 2875	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑m 𝐶))
56		elmapfn 8925	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐶) → 𝑓 Fn 𝐶)
57	55, 56	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
58	57	adantlr 728	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
59	58	3adant3 1150	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
60	53, 1, 54, 59	fsneq 7065	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔‘𝐴) = (𝑓‘𝐴)))
61	51, 60	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
62		simp2 1155	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 ∈ 𝑋)
63	61, 62	eqeltrd 2864	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 ∈ 𝑋)
64	63	3exp 1137	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
65	7, 32, 64	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
66	65	rexlimdv 3165	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → (∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋))
67	30, 66	mpd 16	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 ∈ 𝑋)
68	6, 67	eqelssd 3934	1 ⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑m 𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  {csn 4588  ∪ cuni 4872  ∪ ciun 4956  ran crn 5659   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ↑_m cmap 8885

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887

This theorem is used by: (None)