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Theorem unirnmap 40045
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 (𝜑𝐴𝑉)
unirnmap.x (𝜑𝑋 ⊆ (𝐵𝑚 𝐴))
Assertion
Ref Expression
unirnmap (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐴))

Proof of Theorem unirnmap
Dummy variables 𝑔 𝑥 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unirnmap.x . . . . . . . 8 (𝜑𝑋 ⊆ (𝐵𝑚 𝐴))
21sselda 3761 . . . . . . 7 ((𝜑𝑔𝑋) → 𝑔 ∈ (𝐵𝑚 𝐴))
3 elmapfn 8083 . . . . . . 7 (𝑔 ∈ (𝐵𝑚 𝐴) → 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑔𝑋) → 𝑔 Fn 𝐴)
5 simplr 785 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → 𝑔𝑋)
6 dffn3 6234 . . . . . . . . . . . 12 (𝑔 Fn 𝐴𝑔:𝐴⟶ran 𝑔)
74, 6sylib 209 . . . . . . . . . . 11 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑔)
87ffvelrnda 6549 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
9 rneq 5519 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
109eleq2d 2830 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑔𝑥) ∈ ran 𝑓 ↔ (𝑔𝑥) ∈ ran 𝑔))
1110rspcev 3461 . . . . . . . . . 10 ((𝑔𝑋 ∧ (𝑔𝑥) ∈ ran 𝑔) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
125, 8, 11syl2anc 579 . . . . . . . . 9 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
13 eliun 4680 . . . . . . . . 9 ((𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
1412, 13sylibr 225 . . . . . . . 8 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓)
15 rnuni 5727 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
1614, 15syl6eleqr 2855 . . . . . . 7 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑋)
1716ralrimiva 3113 . . . . . 6 ((𝜑𝑔𝑋) → ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋)
184, 17jca 507 . . . . 5 ((𝜑𝑔𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
19 ffnfv 6578 . . . . 5 (𝑔:𝐴⟶ran 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
2018, 19sylibr 225 . . . 4 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑋)
21 ovexd 6876 . . . . . . . . 9 (𝜑 → (𝐵𝑚 𝐴) ∈ V)
2221, 1ssexd 4966 . . . . . . . 8 (𝜑𝑋 ∈ V)
23 uniexg 7153 . . . . . . . 8 (𝑋 ∈ V → 𝑋 ∈ V)
2422, 23syl 17 . . . . . . 7 (𝜑 𝑋 ∈ V)
25 rnexg 7296 . . . . . . 7 ( 𝑋 ∈ V → ran 𝑋 ∈ V)
2624, 25syl 17 . . . . . 6 (𝜑 → ran 𝑋 ∈ V)
27 unirnmap.a . . . . . 6 (𝜑𝐴𝑉)
2826, 27elmapd 8074 . . . . 5 (𝜑 → (𝑔 ∈ (ran 𝑋𝑚 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2928adantr 472 . . . 4 ((𝜑𝑔𝑋) → (𝑔 ∈ (ran 𝑋𝑚 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
3020, 29mpbird 248 . . 3 ((𝜑𝑔𝑋) → 𝑔 ∈ (ran 𝑋𝑚 𝐴))
3130ralrimiva 3113 . 2 (𝜑 → ∀𝑔𝑋 𝑔 ∈ (ran 𝑋𝑚 𝐴))
32 dfss3 3750 . 2 (𝑋 ⊆ (ran 𝑋𝑚 𝐴) ↔ ∀𝑔𝑋 𝑔 ∈ (ran 𝑋𝑚 𝐴))
3331, 32sylibr 225 1 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  wss 3732   cuni 4594   ciun 4676  ran crn 5278   Fn wfn 6063  wf 6064  cfv 6068  (class class class)co 6842  𝑚 cmap 8060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-map 8062
This theorem is referenced by:  unirnmapsn  40051
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