Theorem unirnmapsn 38401

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 4835	. . . . 5 ⊢ {𝐴} ∈ V
3	1, 2	eqeltri 2684	. . . 4 ⊢ 𝐶 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		unirnmapsn.x	. . 3 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 𝐶))
6	4, 5	unirnmap 38395	. 2 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑𝑚 𝐶))
7		simpl 472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝜑)
8		equid 1926	. . . . . . . . 9 ⊢ 𝑔 = 𝑔
9		rnuni 5463	. . . . . . . . . 10 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
10	9	oveq1i 6559	. . . . . . . . 9 ⊢ (ran ∪ 𝑋 ↑𝑚 𝐶) = (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)
11	8, 10	eleq12i 2681	. . . . . . . 8 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) ↔ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
12	11	biimpi 205	. . . . . . 7 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
13	12	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
14		ovex 6577	. . . . . . . . . . . . 13 ⊢ (𝐵 ↑𝑚 𝐶) ∈ V
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ↑𝑚 𝐶) ∈ V)
16	15, 5	ssexd 4733	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ V)
17		rnexg 6990	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑋 → ran 𝑓 ∈ V)
18	17	rgen 2906	. . . . . . . . . . . 12 ⊢ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
20		iunexg 7035	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
21	16, 19, 20	syl2anc 691	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
22	21, 4	elmapd 7758	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶) ↔ 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓))
23	22	biimpa 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓)
24		unirnmapsn.A	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
25		snidg 4153	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
27	26, 1	syl6eleqr 2699	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
28	27	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝐴 ∈ 𝐶)
29	23, 28	ffvelrnd 6268	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → (𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
30		eliun 4460	. . . . . . 7 ⊢ ((𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
31	29, 30	sylib 207	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
32	7, 13, 31	syl2anc 691	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
33		elmapfn 7766	. . . . . . . 8 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) → 𝑔 Fn 𝐶)
34	33	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 Fn 𝐶)
35		simp3 1056	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ ran 𝑓)
36	24	3ad2ant1 1075	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
37	1	oveq2i 6560	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ↑𝑚 𝐶) = (𝐵 ↑𝑚 {𝐴})
38	5, 37	syl6sseq 3614	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴}))
39	38	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴}))
40		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
41	39, 40	sseldd 3569	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 {𝐴}))
42		unirnmapsn.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
43	42	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐵 ∈ 𝑊)
44	2	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝐴} ∈ V)
45	43, 44	elmapd 7758	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ∈ (𝐵 ↑𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
46	41, 45	mpbid 221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶𝐵)
47	46	3adant3 1074	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
48	36, 47	rnsnf 38365	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓‘𝐴)})
49	35, 48	eleqtrd 2690	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ {(𝑓‘𝐴)})
50		fvex 6113	. . . . . . . . . . . . 13 ⊢ (𝑔‘𝐴) ∈ V
51	50	elsn 4140	. . . . . . . . . . . 12 ⊢ ((𝑔‘𝐴) ∈ {(𝑓‘𝐴)} ↔ (𝑔‘𝐴) = (𝑓‘𝐴))
52	49, 51	sylib 207	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
53	52	3adant1r 1311	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
54	24	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → 𝐴 ∈ 𝑉)
55	54	3ad2ant1 1075	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
56		simp1r 1079	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
57	41, 37	syl6eleqr 2699	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 𝐶))
58		elmapfn 7766	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐶) → 𝑓 Fn 𝐶)
59	57, 58	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
60	59	adantlr 747	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
61	60	3adant3 1074	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
62	55, 1, 56, 61	fsneq 38393	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔‘𝐴) = (𝑓‘𝐴)))
63	53, 62	mpbird 246	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
64		simp2 1055	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 ∈ 𝑋)
65	63, 64	eqeltrd 2688	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 ∈ 𝑋)
66	65	3exp 1256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
67	7, 34, 66	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
68	67	rexlimdv 3012	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → (∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋))
69	32, 68	mpd 15	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 ∈ 𝑋)
70	69	ralrimiva 2949	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)𝑔 ∈ 𝑋)
71		dfss3 3558	. . 3 ⊢ ((ran ∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)𝑔 ∈ 𝑋)
72	70, 71	sylibr 223	. 2 ⊢ (𝜑 → (ran ∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋)
73	6, 72	eqssd 3585	1 ⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑𝑚 𝐶))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {csn 4125  ∪ cuni 4372  ∪ ciun 4455  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↑_𝑚 cmap 7744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-map 7746

This theorem is referenced by: (None)