Theorem unirnmap 42748

Description: Given a subset of a set exponentiation, the base set can be restricted. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
unirnmap.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
unirnmap.x	⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m 𝐴))

Assertion

Ref	Expression
unirnmap	⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐴))

Proof of Theorem unirnmap

Dummy variables 𝑔 𝑥 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		unirnmap.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ (𝐵 ↑m 𝐴))
2	1	sselda 3921	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (𝐵 ↑m 𝐴))
3		elmapfn 8653	. . . . . . 7 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐴) → 𝑔 Fn 𝐴)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 Fn 𝐴)
5		simplr 766	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → 𝑔 ∈ 𝑋)
6		dffn3 6613	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶ran 𝑔)
7	4, 6	sylib 217	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran 𝑔)
8	7	ffvelrnda 6961	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
9		rneq 5845	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
10	9	eleq2d 2824	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑔‘𝑥) ∈ ran 𝑓 ↔ (𝑔‘𝑥) ∈ ran 𝑔))
11	10	rspcev 3561	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝑋 ∧ (𝑔‘𝑥) ∈ ran 𝑔) → ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
12	5, 8, 11	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
13		eliun 4928	. . . . . . . . 9 ⊢ ((𝑔‘𝑥) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
14	12, 13	sylibr 233	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
15		rnuni 6052	. . . . . . . 8 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
16	14, 15	eleqtrrdi 2850	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran ∪ 𝑋)
17	16	ralrimiva 3103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋)
18	4, 17	jca 512	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋))
19		ffnfv 6992	. . . . 5 ⊢ (𝑔:𝐴⟶ran ∪ 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋))
20	18, 19	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran ∪ 𝑋)
21		ovexd 7310	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ∈ V)
22	21, 1	ssexd 5248	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
23	22	uniexd 7595	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑋 ∈ V)
24		rnexg 7751	. . . . . . 7 ⊢ (∪ 𝑋 ∈ V → ran ∪ 𝑋 ∈ V)
25	23, 24	syl 17	. . . . . 6 ⊢ (𝜑 → ran ∪ 𝑋 ∈ V)
26		unirnmap.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
27	25, 26	elmapd 8629	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟶ran ∪ 𝑋))
28	27	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟶ran ∪ 𝑋))
29	20, 28	mpbird 256	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
30	29	ralrimiva 3103	. 2 ⊢ (𝜑 → ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
31		dfss3 3909	. 2 ⊢ (𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐴) ↔ ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
32	30, 31	sylibr 233	1 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐴))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  Vcvv 3432   ⊆ wss 3887  ∪ cuni 4839  ∪ ciun 4924  ran crn 5590   Fn wfn 6428  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275   ↑_m cmap 8615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617

This theorem is referenced by:  unirnmapsn  42754