Theorem unirnmap 42362

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 3887	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (𝐵 ↑m 𝐴))
3		elmapfn 8524	. . . . . . 7 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐴) → 𝑔 Fn 𝐴)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 Fn 𝐴)
5		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → 𝑔 ∈ 𝑋)
6		dffn3 6536	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶ran 𝑔)
7	4, 6	sylib 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran 𝑔)
8	7	ffvelrnda 6882	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
9		rneq 5790	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
10	9	eleq2d 2816	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑔‘𝑥) ∈ ran 𝑓 ↔ (𝑔‘𝑥) ∈ ran 𝑔))
11	10	rspcev 3527	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝑋 ∧ (𝑔‘𝑥) ∈ ran 𝑔) → ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
12	5, 8, 11	syl2anc 587	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
13		eliun 4894	. . . . . . . . 9 ⊢ ((𝑔‘𝑥) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
14	12, 13	sylibr 237	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
15		rnuni 5992	. . . . . . . 8 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
16	14, 15	eleqtrrdi 2842	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran ∪ 𝑋)
17	16	ralrimiva 3095	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋)
18	4, 17	jca 515	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋))
19		ffnfv 6913	. . . . 5 ⊢ (𝑔:𝐴⟶ran ∪ 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋))
20	18, 19	sylibr 237	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran ∪ 𝑋)
21		ovexd 7226	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ∈ V)
22	21, 1	ssexd 5202	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
23	22	uniexd 7508	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑋 ∈ V)
24		rnexg 7660	. . . . . . 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 8500	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟶ran ∪ 𝑋))
28	27	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟶ran ∪ 𝑋))
29	20, 28	mpbird 260	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
30	29	ralrimiva 3095	. 2 ⊢ (𝜑 → ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
31		dfss3 3875	. 2 ⊢ (𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐴) ↔ ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
32	30, 31	sylibr 237	1 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052  Vcvv 3398   ⊆ wss 3853  ∪ cuni 4805  ∪ ciun 4890  ran crn 5537   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7191   ↑_m cmap 8486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-map 8488

This theorem is referenced by:  unirnmapsn  42368