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Theorem unirnmap 44629
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 (πœ‘ β†’ 𝐴 ∈ 𝑉)
unirnmap.x (πœ‘ β†’ 𝑋 βŠ† (𝐡 ↑m 𝐴))
Assertion
Ref Expression
unirnmap (πœ‘ β†’ 𝑋 βŠ† (ran βˆͺ 𝑋 ↑m 𝐴))

Proof of Theorem unirnmap
Dummy variables 𝑔 π‘₯ 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unirnmap.x . . . . . . . 8 (πœ‘ β†’ 𝑋 βŠ† (𝐡 ↑m 𝐴))
21sselda 3982 . . . . . . 7 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔 ∈ (𝐡 ↑m 𝐴))
3 elmapfn 8892 . . . . . . 7 (𝑔 ∈ (𝐡 ↑m 𝐴) β†’ 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔 Fn 𝐴)
5 simplr 767 . . . . . . . . . 10 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ 𝑔 ∈ 𝑋)
6 dffn3 6740 . . . . . . . . . . . 12 (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟢ran 𝑔)
74, 6sylib 217 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔:𝐴⟢ran 𝑔)
87ffvelcdmda 7099 . . . . . . . . . 10 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ (π‘”β€˜π‘₯) ∈ ran 𝑔)
9 rneq 5942 . . . . . . . . . . . 12 (𝑓 = 𝑔 β†’ ran 𝑓 = ran 𝑔)
109eleq2d 2815 . . . . . . . . . . 11 (𝑓 = 𝑔 β†’ ((π‘”β€˜π‘₯) ∈ ran 𝑓 ↔ (π‘”β€˜π‘₯) ∈ ran 𝑔))
1110rspcev 3611 . . . . . . . . . 10 ((𝑔 ∈ 𝑋 ∧ (π‘”β€˜π‘₯) ∈ ran 𝑔) β†’ βˆƒπ‘“ ∈ 𝑋 (π‘”β€˜π‘₯) ∈ ran 𝑓)
125, 8, 11syl2anc 582 . . . . . . . . 9 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ βˆƒπ‘“ ∈ 𝑋 (π‘”β€˜π‘₯) ∈ ran 𝑓)
13 eliun 5004 . . . . . . . . 9 ((π‘”β€˜π‘₯) ∈ βˆͺ 𝑓 ∈ 𝑋 ran 𝑓 ↔ βˆƒπ‘“ ∈ 𝑋 (π‘”β€˜π‘₯) ∈ ran 𝑓)
1412, 13sylibr 233 . . . . . . . 8 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ (π‘”β€˜π‘₯) ∈ βˆͺ 𝑓 ∈ 𝑋 ran 𝑓)
15 rnuni 6158 . . . . . . . 8 ran βˆͺ 𝑋 = βˆͺ 𝑓 ∈ 𝑋 ran 𝑓
1614, 15eleqtrrdi 2840 . . . . . . 7 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ (π‘”β€˜π‘₯) ∈ ran βˆͺ 𝑋)
1716ralrimiva 3143 . . . . . 6 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ βˆ€π‘₯ ∈ 𝐴 (π‘”β€˜π‘₯) ∈ ran βˆͺ 𝑋)
184, 17jca 510 . . . . 5 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ (𝑔 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘”β€˜π‘₯) ∈ ran βˆͺ 𝑋))
19 ffnfv 7134 . . . . 5 (𝑔:𝐴⟢ran βˆͺ 𝑋 ↔ (𝑔 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘”β€˜π‘₯) ∈ ran βˆͺ 𝑋))
2018, 19sylibr 233 . . . 4 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔:𝐴⟢ran βˆͺ 𝑋)
21 ovexd 7461 . . . . . . . . 9 (πœ‘ β†’ (𝐡 ↑m 𝐴) ∈ V)
2221, 1ssexd 5328 . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ V)
2322uniexd 7755 . . . . . . 7 (πœ‘ β†’ βˆͺ 𝑋 ∈ V)
24 rnexg 7918 . . . . . . 7 (βˆͺ 𝑋 ∈ V β†’ ran βˆͺ 𝑋 ∈ V)
2523, 24syl 17 . . . . . 6 (πœ‘ β†’ ran βˆͺ 𝑋 ∈ V)
26 unirnmap.a . . . . . 6 (πœ‘ β†’ 𝐴 ∈ 𝑉)
2725, 26elmapd 8867 . . . . 5 (πœ‘ β†’ (𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟢ran βˆͺ 𝑋))
2827adantr 479 . . . 4 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ (𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟢ran βˆͺ 𝑋))
2920, 28mpbird 256 . . 3 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴))
3029ralrimiva 3143 . 2 (πœ‘ β†’ βˆ€π‘” ∈ 𝑋 𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴))
31 dfss3 3970 . 2 (𝑋 βŠ† (ran βˆͺ 𝑋 ↑m 𝐴) ↔ βˆ€π‘” ∈ 𝑋 𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴))
3230, 31sylibr 233 1 (πœ‘ β†’ 𝑋 βŠ† (ran βˆͺ 𝑋 ↑m 𝐴))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  βˆƒwrex 3067  Vcvv 3473   βŠ† wss 3949  βˆͺ cuni 4912  βˆͺ ciun 5000  ran crn 5683   Fn wfn 6548  βŸΆwf 6549  β€˜cfv 6553  (class class class)co 7426   ↑m cmap 8853
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7748
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 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-map 8855
This theorem is referenced by:  unirnmapsn  44635
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