Theorem unirnmap 45202

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 3946	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (𝐵 ↑m 𝐴))
3		elmapfn 8838	. . . . . . 7 ⊢ (𝑔 ∈ (𝐵 ↑m 𝐴) → 𝑔 Fn 𝐴)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 Fn 𝐴)
5		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → 𝑔 ∈ 𝑋)
6		dffn3 6700	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶ran 𝑔)
7	4, 6	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran 𝑔)
8	7	ffvelcdmda 7056	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
9		rneq 5900	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
10	9	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑔‘𝑥) ∈ ran 𝑓 ↔ (𝑔‘𝑥) ∈ ran 𝑔))
11	10	rspcev 3588	. . . . . . . . . 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 4959	. . . . . . . . 9 ⊢ ((𝑔‘𝑥) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
14	12, 13	sylibr 234	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
15		rnuni 6121	. . . . . . . 8 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
16	14, 15	eleqtrrdi 2839	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran ∪ 𝑋)
17	16	ralrimiva 3125	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋)
18	4, 17	jca 511	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋))
19		ffnfv 7091	. . . . 5 ⊢ (𝑔:𝐴⟶ran ∪ 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋))
20	18, 19	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran ∪ 𝑋)
21		ovexd 7422	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ∈ V)
22	21, 1	ssexd 5279	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
23	22	uniexd 7718	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑋 ∈ V)
24		rnexg 7878	. . . . . . 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 8813	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟶ran ∪ 𝑋))
28	27	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → (𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟶ran ∪ 𝑋))
29	20, 28	mpbird 257	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
30	29	ralrimiva 3125	. 2 ⊢ (𝜑 → ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
31		dfss3 3935	. 2 ⊢ (𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐴) ↔ ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
32	30, 31	sylibr 234	1 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑m 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ⊆ wss 3914  ∪ cuni 4871  ∪ ciun 4955  ran crn 5639   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387   ↑_m cmap 8799

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-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

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-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801

This theorem is referenced by:  unirnmapsn  45208