Theorem unirnmap 39919

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 3753	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (𝐵 ↑𝑚 𝐴))
3		elmapfn 8033	. . . . . . 7 ⊢ (𝑔 ∈ (𝐵 ↑𝑚 𝐴) → 𝑔 Fn 𝐴)
4	2, 3	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 Fn 𝐴)
5		simplr 746	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → 𝑔 ∈ 𝑋)
6		dffn3 6195	. . . . . . . . . . . 12 ⊢ (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟶ran 𝑔)
7	4, 6	sylib 208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran 𝑔)
8	7	ffvelrnda 6503	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran 𝑔)
9		rneq 5490	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
10	9	eleq2d 2836	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → ((𝑔‘𝑥) ∈ ran 𝑓 ↔ (𝑔‘𝑥) ∈ ran 𝑔))
11	10	rspcev 3461	. . . . . . . . . 10 ⊢ ((𝑔 ∈ 𝑋 ∧ (𝑔‘𝑥) ∈ ran 𝑔) → ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
12	5, 8, 11	syl2anc 567	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
13		eliun 4659	. . . . . . . . 9 ⊢ ((𝑔‘𝑥) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓 ↔ ∃𝑓 ∈ 𝑋 (𝑔‘𝑥) ∈ ran 𝑓)
14	12, 13	sylibr 224	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ∪ 𝑓 ∈ 𝑋 ran 𝑓)
15		rnuni 5686	. . . . . . . 8 ⊢ ran ∪ 𝑋 = ∪ 𝑓 ∈ 𝑋 ran 𝑓
16	14, 15	syl6eleqr 2861	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 ∈ 𝑋) ∧ 𝑥 ∈ 𝐴) → (𝑔‘𝑥) ∈ ran ∪ 𝑋)
17	16	ralrimiva 3115	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋)
18	4, 17	jca 497	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋))
19		ffnfv 6531	. . . . 5 ⊢ (𝑔:𝐴⟶ran ∪ 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑔‘𝑥) ∈ ran ∪ 𝑋))
20	18, 19	sylibr 224	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔:𝐴⟶ran ∪ 𝑋)
21		ovexd 6826	. . . . . . . . 9 ⊢ (𝜑 → (𝐵 ↑𝑚 𝐴) ∈ V)
22	21, 1	ssexd 4940	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ V)
23		uniexg 7103	. . . . . . . 8 ⊢ (𝑋 ∈ V → ∪ 𝑋 ∈ V)
24	22, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → ∪ 𝑋 ∈ V)
25		rnexg 7246	. . . . . . 7 ⊢ (∪ 𝑋 ∈ V → ran ∪ 𝑋 ∈ V)
26	24, 25	syl 17	. . . . . 6 ⊢ (𝜑 → ran ∪ 𝑋 ∈ V)
27		unirnmap.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
28	26, 27	elmapd 8024	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐴) ↔ 𝑔:𝐴⟶ran ∪ 𝑋))
29	28	adantr 466	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → (𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐴) ↔ 𝑔:𝐴⟶ran ∪ 𝑋))
30	20, 29	mpbird 247	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑋) → 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐴))
31	30	ralrimiva 3115	. 2 ⊢ (𝜑 → ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐴))
32		dfss3 3742	. 2 ⊢ (𝑋 ⊆ (ran ∪ 𝑋 ↑𝑚 𝐴) ↔ ∀𝑔 ∈ 𝑋 𝑔 ∈ (ran ∪ 𝑋 ↑𝑚 𝐴))
33	31, 32	sylibr 224	1 ⊢ (𝜑 → 𝑋 ⊆ (ran ∪ 𝑋 ↑𝑚 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ⊆ wss 3724  ∪ cuni 4575  ∪ ciun 4655  ran crn 5251   Fn wfn 6027  ⟶wf 6028  ‘cfv 6032  (class class class)co 6794   ↑_𝑚 cmap 8010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7097

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-1st 7316  df-2nd 7317  df-map 8012

This theorem is referenced by:  unirnmapsn  39925