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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 (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 3753 . . . . . . 7 ((𝜑𝑔𝑋) → 𝑔 ∈ (𝐵𝑚 𝐴))
3 elmapfn 8033 . . . . . . 7 (𝑔 ∈ (𝐵𝑚 𝐴) → 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑔𝑋) → 𝑔 Fn 𝐴)
5 simplr 746 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → 𝑔𝑋)
6 dffn3 6195 . . . . . . . . . . . 12 (𝑔 Fn 𝐴𝑔:𝐴⟶ran 𝑔)
74, 6sylib 208 . . . . . . . . . . 11 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑔)
87ffvelrnda 6503 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
9 rneq 5490 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
109eleq2d 2836 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑔𝑥) ∈ ran 𝑓 ↔ (𝑔𝑥) ∈ ran 𝑔))
1110rspcev 3461 . . . . . . . . . 10 ((𝑔𝑋 ∧ (𝑔𝑥) ∈ ran 𝑔) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
125, 8, 11syl2anc 567 . . . . . . . . 9 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
13 eliun 4659 . . . . . . . . 9 ((𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
1412, 13sylibr 224 . . . . . . . 8 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓)
15 rnuni 5686 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
1614, 15syl6eleqr 2861 . . . . . . 7 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑋)
1716ralrimiva 3115 . . . . . 6 ((𝜑𝑔𝑋) → ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋)
184, 17jca 497 . . . . 5 ((𝜑𝑔𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
19 ffnfv 6531 . . . . 5 (𝑔:𝐴⟶ran 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
2018, 19sylibr 224 . . . 4 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑋)
21 ovexd 6826 . . . . . . . . 9 (𝜑 → (𝐵𝑚 𝐴) ∈ V)
2221, 1ssexd 4940 . . . . . . . 8 (𝜑𝑋 ∈ V)
23 uniexg 7103 . . . . . . . 8 (𝑋 ∈ V → 𝑋 ∈ V)
2422, 23syl 17 . . . . . . 7 (𝜑 𝑋 ∈ V)
25 rnexg 7246 . . . . . . 7 ( 𝑋 ∈ V → ran 𝑋 ∈ V)
2624, 25syl 17 . . . . . 6 (𝜑 → ran 𝑋 ∈ V)
27 unirnmap.a . . . . . 6 (𝜑𝐴𝑉)
2826, 27elmapd 8024 . . . . 5 (𝜑 → (𝑔 ∈ (ran 𝑋𝑚 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2928adantr 466 . . . 4 ((𝜑𝑔𝑋) → (𝑔 ∈ (ran 𝑋𝑚 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
3020, 29mpbird 247 . . 3 ((𝜑𝑔𝑋) → 𝑔 ∈ (ran 𝑋𝑚 𝐴))
3130ralrimiva 3115 . 2 (𝜑 → ∀𝑔𝑋 𝑔 ∈ (ran 𝑋𝑚 𝐴))
32 dfss3 3742 . 2 (𝑋 ⊆ (ran 𝑋𝑚 𝐴) ↔ ∀𝑔𝑋 𝑔 ∈ (ran 𝑋𝑚 𝐴))
3331, 32sylibr 224 1 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐴))
Colors of variables: wff setvar class
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
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