Theorem unirnmapsn 45196

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 5375	. . . . 5 ⊢ {𝐴} ∈ V
3	1, 2	eqeltri 2824	. . . 4 ⊢ 𝐶 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		unirnmapsn.x	. . 3 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m 𝐶))
6	4, 5	unirnmap 45190	. 2 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐶))
7		simpl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝜑)
8		equid 2012	. . . . . . 7 ⊢ 𝑔 = 𝑔
9		rnuni 6097	. . . . . . . 8 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
10	9	oveq1i 7359	. . . . . . 7 ⊢ (ran ∪ 𝑋 ↑m 𝐶) = (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)
11	8, 10	eleq12i 2821	. . . . . 6 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶) ↔ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶))
12	11	biimpi 216	. . . . 5 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶))
13	12	adantl 481	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶))
14		ovexd 7384	. . . . . . . . . 10 ⊢ (𝜑 → (𝐵 ↑m 𝐶) ∈ V)
15	14, 5	ssexd 5263	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ V)
16		rnexg 7835	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝑋 → ran 𝑓 ∈ V)
17	16	rgen 3046	. . . . . . . . . 10 ⊢ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V
18	17	a1i 11	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
19		iunexg 7898	. . . . . . . . 9 ⊢ ((𝑋 ∈ V ∧ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
20	15, 18, 19	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
21	20, 4	elmapd 8767	. . . . . . 7 ⊢ (𝜑 → (𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶) ↔ 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓))
22	21	biimpa 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓)
23		unirnmapsn.A	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
24		snidg 4612	. . . . . . . . 9 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
25	23, 24	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
26	25, 1	eleqtrrdi 2839	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
27	26	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → 𝐴 ∈ 𝐶)
28	22, 27	ffvelcdmd 7019	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → (𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
29		eliun 4945	. . . . 5 ⊢ ((𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
30	28, 29	sylib 218	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑m 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
31	7, 13, 30	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
32		elmapfn 8792	. . . . . 6 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶) → 𝑔 Fn 𝐶)
33	32	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 Fn 𝐶)
34		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ ran 𝑓)
35	23	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
36	1	oveq2i 7360	. . . . . . . . . . . . . . . . 17 ⊢ (𝐵 ↑m 𝐶) = (𝐵 ↑m {𝐴})
37	5, 36	sseqtrdi 3976	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m {𝐴}))
38	37	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (𝐵 ↑m {𝐴}))
39		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
40	38, 39	sseldd 3936	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑m {𝐴}))
41		unirnmapsn.b	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
42	41	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐵 ∈ 𝑊)
43	2	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝐴} ∈ V)
44	42, 43	elmapd 8767	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ∈ (𝐵 ↑m {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
45	40, 44	mpbid 232	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶𝐵)
46	45	3adant3 1132	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
47	35, 46	rnsnf 45166	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓‘𝐴)})
48	34, 47	eleqtrd 2830	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ {(𝑓‘𝐴)})
49		fvex 6835	. . . . . . . . . . 11 ⊢ (𝑔‘𝐴) ∈ V
50	49	elsn 4592	. . . . . . . . . 10 ⊢ ((𝑔‘𝐴) ∈ {(𝑓‘𝐴)} ↔ (𝑔‘𝐴) = (𝑓‘𝐴))
51	48, 50	sylib 218	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
52	51	3adant1r 1178	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
53	23	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → 𝐴 ∈ 𝑉)
54	53	3ad2ant1 1133	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
55		simp1r 1199	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
56	40, 36	eleqtrrdi 2839	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑m 𝐶))
57		elmapfn 8792	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ (𝐵 ↑m 𝐶) → 𝑓 Fn 𝐶)
58	56, 57	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
59	58	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
60	59	3adant3 1132	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
61	54, 1, 55, 60	fsneq 45188	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔‘𝐴) = (𝑓‘𝐴)))
62	52, 61	mpbird 257	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
63		simp2 1137	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 ∈ 𝑋)
64	62, 63	eqeltrd 2828	. . . . . 6 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 ∈ 𝑋)
65	64	3exp 1119	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
66	7, 33, 65	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
67	66	rexlimdv 3128	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → (∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋))
68	31, 67	mpd 15	. 2 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐶)) → 𝑔 ∈ 𝑋)
69	6, 68	eqelssd 3957	1 ⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑m 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3436   ⊆ wss 3903  {csn 4577  ∪ cuni 4858  ∪ ciun 4941  ran crn 5620   Fn wfn 6477  ⟶wf 6478  ‘cfv 6482  (class class class)co 7349   ↑_m cmap 8753

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-map 8755

This theorem is referenced by: (None)