Theorem unirnmap 45251

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 3934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (𝐵 ↑m 𝐴))
3		elmapfn 8789	. . . . . . 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 6663	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶ran 𝑔)
7	4, 6	sylib 218	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran 𝑔)
8	7	ffvelcdmda 7017	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
9		rneq 5876	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
10	9	eleq2d 2817	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑔‘𝑥) ∈ ran 𝑓 ↔ (𝑔‘𝑥) ∈ ran 𝑔))
11	10	rspcev 3577	. . . . . . . . . 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 4945	. . . . . . . . 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 6095	. . . . . . . 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 3124	. . . . . 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 7052	. . . . 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 7381	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↑m 𝐴) ∈ V)
22	21, 1	ssexd 5262	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
23	22	uniexd 7675	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑋 ∈ V)
24		rnexg 7832	. . . . . . 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 8764	. . . . 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 3124	. 2 ⊢ (𝜑 → ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑m 𝐴))
31		dfss3 3923	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ⊆ wss 3902  ∪ cuni 4859  ∪ ciun 4941  ran crn 5617   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-map 8752

This theorem is referenced by:  unirnmapsn  45257