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Theorem unirnmap 40875
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 3857 . . . . . . 7 ((𝜑𝑔𝑋) → 𝑔 ∈ (𝐵𝑚 𝐴))
3 elmapfn 8225 . . . . . . 7 (𝑔 ∈ (𝐵𝑚 𝐴) → 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑔𝑋) → 𝑔 Fn 𝐴)
5 simplr 756 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → 𝑔𝑋)
6 dffn3 6353 . . . . . . . . . . . 12 (𝑔 Fn 𝐴𝑔:𝐴⟶ran 𝑔)
74, 6sylib 210 . . . . . . . . . . 11 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑔)
87ffvelrnda 6674 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
9 rneq 5646 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
109eleq2d 2848 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑔𝑥) ∈ ran 𝑓 ↔ (𝑔𝑥) ∈ ran 𝑔))
1110rspcev 3532 . . . . . . . . . 10 ((𝑔𝑋 ∧ (𝑔𝑥) ∈ ran 𝑔) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
125, 8, 11syl2anc 576 . . . . . . . . 9 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
13 eliun 4794 . . . . . . . . 9 ((𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
1412, 13sylibr 226 . . . . . . . 8 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓)
15 rnuni 5845 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
1614, 15syl6eleqr 2874 . . . . . . 7 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑋)
1716ralrimiva 3129 . . . . . 6 ((𝜑𝑔𝑋) → ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋)
184, 17jca 504 . . . . 5 ((𝜑𝑔𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
19 ffnfv 6703 . . . . 5 (𝑔:𝐴⟶ran 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
2018, 19sylibr 226 . . . 4 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑋)
21 ovexd 7008 . . . . . . . . 9 (𝜑 → (𝐵𝑚 𝐴) ∈ V)
2221, 1ssexd 5082 . . . . . . . 8 (𝜑𝑋 ∈ V)
23 uniexg 7283 . . . . . . . 8 (𝑋 ∈ V → 𝑋 ∈ V)
2422, 23syl 17 . . . . . . 7 (𝜑 𝑋 ∈ V)
25 rnexg 7427 . . . . . . 7 ( 𝑋 ∈ V → ran 𝑋 ∈ V)
2624, 25syl 17 . . . . . 6 (𝜑 → ran 𝑋 ∈ V)
27 unirnmap.a . . . . . 6 (𝜑𝐴𝑉)
2826, 27elmapd 8216 . . . . 5 (𝜑 → (𝑔 ∈ (ran 𝑋𝑚 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2928adantr 473 . . . 4 ((𝜑𝑔𝑋) → (𝑔 ∈ (ran 𝑋𝑚 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
3020, 29mpbird 249 . . 3 ((𝜑𝑔𝑋) → 𝑔 ∈ (ran 𝑋𝑚 𝐴))
3130ralrimiva 3129 . 2 (𝜑 → ∀𝑔𝑋 𝑔 ∈ (ran 𝑋𝑚 𝐴))
32 dfss3 3846 . 2 (𝑋 ⊆ (ran 𝑋𝑚 𝐴) ↔ ∀𝑔𝑋 𝑔 ∈ (ran 𝑋𝑚 𝐴))
3331, 32sylibr 226 1 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  wral 3085  wrex 3086  Vcvv 3412  wss 3828   cuni 4710   ciun 4790  ran crn 5405   Fn wfn 6181  wf 6182  cfv 6186  (class class class)co 6974  𝑚 cmap 8202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2747  ax-sep 5058  ax-nul 5065  ax-pow 5117  ax-pr 5184  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2756  df-cleq 2768  df-clel 2843  df-nfc 2915  df-ne 2965  df-ral 3090  df-rex 3091  df-rab 3094  df-v 3414  df-sbc 3681  df-csb 3786  df-dif 3831  df-un 3833  df-in 3835  df-ss 3842  df-nul 4178  df-if 4349  df-pw 4422  df-sn 4440  df-pr 4442  df-op 4446  df-uni 4711  df-iun 4792  df-br 4928  df-opab 4990  df-mpt 5007  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-fv 6194  df-ov 6977  df-oprab 6978  df-mpo 6979  df-1st 7498  df-2nd 7499  df-map 8204
This theorem is referenced by:  unirnmapsn  40881
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