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Theorem unirnmap 43075
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 3932 . . . . . . 7 ((𝜑𝑔𝑋) → 𝑔 ∈ (𝐵m 𝐴))
3 elmapfn 8724 . . . . . . 7 (𝑔 ∈ (𝐵m 𝐴) → 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑔𝑋) → 𝑔 Fn 𝐴)
5 simplr 766 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → 𝑔𝑋)
6 dffn3 6664 . . . . . . . . . . . 12 (𝑔 Fn 𝐴𝑔:𝐴⟶ran 𝑔)
74, 6sylib 217 . . . . . . . . . . 11 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑔)
87ffvelcdmda 7017 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
9 rneq 5877 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
109eleq2d 2822 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑔𝑥) ∈ ran 𝑓 ↔ (𝑔𝑥) ∈ ran 𝑔))
1110rspcev 3570 . . . . . . . . . 10 ((𝑔𝑋 ∧ (𝑔𝑥) ∈ ran 𝑔) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
125, 8, 11syl2anc 584 . . . . . . . . 9 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
13 eliun 4945 . . . . . . . . 9 ((𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
1412, 13sylibr 233 . . . . . . . 8 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓)
15 rnuni 6087 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
1614, 15eleqtrrdi 2848 . . . . . . 7 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑋)
1716ralrimiva 3139 . . . . . 6 ((𝜑𝑔𝑋) → ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋)
184, 17jca 512 . . . . 5 ((𝜑𝑔𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
19 ffnfv 7048 . . . . 5 (𝑔:𝐴⟶ran 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
2018, 19sylibr 233 . . . 4 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑋)
21 ovexd 7372 . . . . . . . . 9 (𝜑 → (𝐵m 𝐴) ∈ V)
2221, 1ssexd 5268 . . . . . . . 8 (𝜑𝑋 ∈ V)
2322uniexd 7657 . . . . . . 7 (𝜑 𝑋 ∈ V)
24 rnexg 7819 . . . . . . 7 ( 𝑋 ∈ V → ran 𝑋 ∈ V)
2523, 24syl 17 . . . . . 6 (𝜑 → ran 𝑋 ∈ V)
26 unirnmap.a . . . . . 6 (𝜑𝐴𝑉)
2725, 26elmapd 8700 . . . . 5 (𝜑 → (𝑔 ∈ (ran 𝑋m 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2827adantr 481 . . . 4 ((𝜑𝑔𝑋) → (𝑔 ∈ (ran 𝑋m 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2920, 28mpbird 256 . . 3 ((𝜑𝑔𝑋) → 𝑔 ∈ (ran 𝑋m 𝐴))
3029ralrimiva 3139 . 2 (𝜑 → ∀𝑔𝑋 𝑔 ∈ (ran 𝑋m 𝐴))
31 dfss3 3920 . 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 396   = wceq 1540  wcel 2105  wral 3061  wrex 3070  Vcvv 3441  wss 3898   cuni 4852   ciun 4941  ran crn 5621   Fn wfn 6474  wf 6475  cfv 6479  (class class class)co 7337  m cmap 8686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-map 8688
This theorem is referenced by:  unirnmapsn  43081
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