Theorem unirnmapsn 38880

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 4869	. . . . 5 ⊢ {𝐴} ∈ V
3	1, 2	eqeltri 2694	. . . 4 ⊢ 𝐶 ∈ V
4	3	a1i 11	. . 3 ⊢ (𝜑 → 𝐶 ∈ V)
5		unirnmapsn.x	. . 3 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 𝐶))
6	4, 5	unirnmap 38874	. 2 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑𝑚 𝐶))
7		simpl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝜑)
8		equid 1936	. . . . . . . . 9 ⊢ 𝑔 = 𝑔
9		rnuni 5503	. . . . . . . . . 10 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
10	9	oveq1i 6614	. . . . . . . . 9 ⊢ (ran ∪ 𝑋 ↑𝑚 𝐶) = (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)
11	8, 10	eleq12i 2691	. . . . . . . 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 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶))
14		ovex 6632	. . . . . . . . . . . . 13 ⊢ (𝐵 ↑𝑚 𝐶) ∈ V
15	14	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 ↑𝑚 𝐶) ∈ V)
16	15, 5	ssexd 4765	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ V)
17		rnexg 7045	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝑋 → ran 𝑓 ∈ V)
18	17	rgen 2917	. . . . . . . . . . . 12 ⊢ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V
19	18	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
20		iunexg 7089	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ V ∧ ∀𝑓 ∈ 𝑋 ran 𝑓 ∈ V) → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
21	16, 19, 20	syl2anc 692	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑓 ∈ 𝑋 ran 𝑓 ∈ V)
22	21, 4	elmapd 7816	. . . . . . . . 9 ⊢ (𝜑 → (𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶) ↔ 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓))
23	22	biimpa 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝑔:𝐶⟶∪ 𝑓 ∈ 𝑋 ran 𝑓)
24		unirnmapsn.A	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
25		snidg 4177	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
26	24, 25	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
27	26, 1	syl6eleqr 2709	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ 𝐶)
28	27	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → 𝐴 ∈ 𝐶)
29	23, 28	ffvelrnd 6316	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → (𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
30		eliun 4490	. . . . . . 7 ⊢ ((𝑔‘𝐴) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
31	29, 30	sylib 208	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (∪ 𝑓 ∈ 𝑋 ran 𝑓 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
32	7, 13, 31	syl2anc 692	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → ∃𝑓 ∈ 𝑋 (𝑔‘𝐴) ∈ ran 𝑓)
33		elmapfn 7824	. . . . . . . 8 ⊢ (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶) → 𝑔 Fn 𝐶)
34	33	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → 𝑔 Fn 𝐶)
35		simp3 1061	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ ran 𝑓)
36	24	3ad2ant1 1080	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
37	1	oveq2i 6615	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ↑𝑚 𝐶) = (𝐵 ↑𝑚 {𝐴})
38	5, 37	syl6sseq 3630	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴}))
39	38	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑋 ⊆ (𝐵 ↑𝑚 {𝐴}))
40		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ 𝑋)
41	39, 40	sseldd 3584	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 {𝐴}))
42		unirnmapsn.b	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
43	42	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝐵 ∈ 𝑊)
44	2	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → {𝐴} ∈ V)
45	43, 44	elmapd 7816	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝑓 ∈ (𝐵 ↑𝑚 {𝐴}) ↔ 𝑓:{𝐴}⟶𝐵))
46	41, 45	mpbid 222	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓:{𝐴}⟶𝐵)
47	46	3adant3 1079	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓:{𝐴}⟶𝐵)
48	36, 47	rnsnf 38844	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → ran 𝑓 = {(𝑓‘𝐴)})
49	35, 48	eleqtrd 2700	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) ∈ {(𝑓‘𝐴)})
50		fvex 6158	. . . . . . . . . . . . 13 ⊢ (𝑔‘𝐴) ∈ V
51	50	elsn 4163	. . . . . . . . . . . 12 ⊢ ((𝑔‘𝐴) ∈ {(𝑓‘𝐴)} ↔ (𝑔‘𝐴) = (𝑓‘𝐴))
52	49, 51	sylib 208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
53	52	3adant1r 1316	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔‘𝐴) = (𝑓‘𝐴))
54	24	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → 𝐴 ∈ 𝑉)
55	54	3ad2ant1 1080	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝐴 ∈ 𝑉)
56		simp1r 1084	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 Fn 𝐶)
57	41, 37	syl6eleqr 2709	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 ∈ (𝐵 ↑𝑚 𝐶))
58		elmapfn 7824	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ (𝐵 ↑𝑚 𝐶) → 𝑓 Fn 𝐶)
59	57, 58	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
60	59	adantlr 750	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋) → 𝑓 Fn 𝐶)
61	60	3adant3 1079	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 Fn 𝐶)
62	55, 1, 56, 61	fsneq 38872	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → (𝑔 = 𝑓 ↔ (𝑔‘𝐴) = (𝑓‘𝐴)))
63	53, 62	mpbird 247	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 = 𝑓)
64		simp2 1060	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑓 ∈ 𝑋)
65	63, 64	eqeltrd 2698	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 Fn 𝐶) ∧ 𝑓 ∈ 𝑋 ∧ (𝑔‘𝐴) ∈ ran 𝑓) → 𝑔 ∈ 𝑋)
66	65	3exp 1261	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 Fn 𝐶) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
67	7, 34, 66	syl2anc 692	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)) → (𝑓 ∈ 𝑋 → ((𝑔‘𝐴) ∈ ran 𝑓 → 𝑔 ∈ 𝑋)))
68	67	rexlimdv 3023	. . . . 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 2960	. . 3 ⊢ (𝜑 → ∀𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)𝑔 ∈ 𝑋)
71		dfss3 3573	. . 3 ⊢ ((ran ∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋 ↔ ∀𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐶)𝑔 ∈ 𝑋)
72	70, 71	sylibr 224	. 2 ⊢ (𝜑 → (ran ∪ 𝑋 ↑𝑚 𝐶) ⊆ 𝑋)
73	6, 72	eqssd 3600	1 ⊢ (𝜑 → 𝑋 = (ran ∪ 𝑋 ↑𝑚 𝐶))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987  ∀wral 2907  ∃wrex 2908  Vcvv 3186   ⊆ wss 3555  {csn 4148  ∪ cuni 4402  ∪ ciun 4485  ran crn 5075   Fn wfn 5842  ⟶wf 5843  ‘cfv 5847  (class class class)co 6604   ↑_𝑚 cmap 7802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804

This theorem is referenced by: (None)