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Theorem unirnmap 43419
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 3944 . . . . . . 7 ((𝜑𝑔𝑋) → 𝑔 ∈ (𝐵m 𝐴))
3 elmapfn 8803 . . . . . . 7 (𝑔 ∈ (𝐵m 𝐴) → 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑔𝑋) → 𝑔 Fn 𝐴)
5 simplr 767 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → 𝑔𝑋)
6 dffn3 6681 . . . . . . . . . . . 12 (𝑔 Fn 𝐴𝑔:𝐴⟶ran 𝑔)
74, 6sylib 217 . . . . . . . . . . 11 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑔)
87ffvelcdmda 7035 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
9 rneq 5891 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
109eleq2d 2823 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑔𝑥) ∈ ran 𝑓 ↔ (𝑔𝑥) ∈ ran 𝑔))
1110rspcev 3581 . . . . . . . . . 10 ((𝑔𝑋 ∧ (𝑔𝑥) ∈ ran 𝑔) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
125, 8, 11syl2anc 584 . . . . . . . . 9 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
13 eliun 4958 . . . . . . . . 9 ((𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
1412, 13sylibr 233 . . . . . . . 8 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓)
15 rnuni 6101 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
1614, 15eleqtrrdi 2849 . . . . . . 7 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑋)
1716ralrimiva 3143 . . . . . 6 ((𝜑𝑔𝑋) → ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋)
184, 17jca 512 . . . . 5 ((𝜑𝑔𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
19 ffnfv 7066 . . . . 5 (𝑔:𝐴⟶ran 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
2018, 19sylibr 233 . . . 4 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑋)
21 ovexd 7392 . . . . . . . . 9 (𝜑 → (𝐵m 𝐴) ∈ V)
2221, 1ssexd 5281 . . . . . . . 8 (𝜑𝑋 ∈ V)
2322uniexd 7679 . . . . . . 7 (𝜑 𝑋 ∈ V)
24 rnexg 7841 . . . . . . 7 ( 𝑋 ∈ V → ran 𝑋 ∈ V)
2523, 24syl 17 . . . . . 6 (𝜑 → ran 𝑋 ∈ V)
26 unirnmap.a . . . . . 6 (𝜑𝐴𝑉)
2725, 26elmapd 8779 . . . . 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 3143 . 2 (𝜑 → ∀𝑔𝑋 𝑔 ∈ (ran 𝑋m 𝐴))
31 dfss3 3932 . 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 1541  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910   cuni 4865   ciun 4954  ran crn 5634   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  m cmap 8765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-map 8767
This theorem is referenced by:  unirnmapsn  43425
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