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Theorem unirnmap 44717
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 3976 . . . . . . 7 ((𝜑𝑔𝑋) → 𝑔 ∈ (𝐵m 𝐴))
3 elmapfn 8884 . . . . . . 7 (𝑔 ∈ (𝐵m 𝐴) → 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑔𝑋) → 𝑔 Fn 𝐴)
5 simplr 767 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → 𝑔𝑋)
6 dffn3 6735 . . . . . . . . . . . 12 (𝑔 Fn 𝐴𝑔:𝐴⟶ran 𝑔)
74, 6sylib 217 . . . . . . . . . . 11 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑔)
87ffvelcdmda 7093 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
9 rneq 5938 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
109eleq2d 2811 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑔𝑥) ∈ ran 𝑓 ↔ (𝑔𝑥) ∈ ran 𝑔))
1110rspcev 3606 . . . . . . . . . 10 ((𝑔𝑋 ∧ (𝑔𝑥) ∈ ran 𝑔) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
125, 8, 11syl2anc 582 . . . . . . . . 9 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
13 eliun 5001 . . . . . . . . 9 ((𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
1412, 13sylibr 233 . . . . . . . 8 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓)
15 rnuni 6155 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
1614, 15eleqtrrdi 2836 . . . . . . 7 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑋)
1716ralrimiva 3135 . . . . . 6 ((𝜑𝑔𝑋) → ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋)
184, 17jca 510 . . . . 5 ((𝜑𝑔𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
19 ffnfv 7128 . . . . 5 (𝑔:𝐴⟶ran 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
2018, 19sylibr 233 . . . 4 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑋)
21 ovexd 7454 . . . . . . . . 9 (𝜑 → (𝐵m 𝐴) ∈ V)
2221, 1ssexd 5325 . . . . . . . 8 (𝜑𝑋 ∈ V)
2322uniexd 7748 . . . . . . 7 (𝜑 𝑋 ∈ V)
24 rnexg 7910 . . . . . . 7 ( 𝑋 ∈ V → ran 𝑋 ∈ V)
2523, 24syl 17 . . . . . 6 (𝜑 → ran 𝑋 ∈ V)
26 unirnmap.a . . . . . 6 (𝜑𝐴𝑉)
2725, 26elmapd 8859 . . . . 5 (𝜑 → (𝑔 ∈ (ran 𝑋m 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2827adantr 479 . . . 4 ((𝜑𝑔𝑋) → (𝑔 ∈ (ran 𝑋m 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2920, 28mpbird 256 . . 3 ((𝜑𝑔𝑋) → 𝑔 ∈ (ran 𝑋m 𝐴))
3029ralrimiva 3135 . 2 (𝜑 → ∀𝑔𝑋 𝑔 ∈ (ran 𝑋m 𝐴))
31 dfss3 3965 . 2 (𝑋 ⊆ (ran 𝑋m 𝐴) ↔ ∀𝑔𝑋 𝑔 ∈ (ran 𝑋m 𝐴))
3230, 31sylibr 233 1 (𝜑𝑋 ⊆ (ran 𝑋m 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  Vcvv 3461  wss 3944   cuni 4909   ciun 4997  ran crn 5679   Fn wfn 6544  wf 6545  cfv 6549  (class class class)co 7419  m cmap 8845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-map 8847
This theorem is referenced by:  unirnmapsn  44723
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