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Theorem unirnmap 44479
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 3977 . . . . . . 7 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔 ∈ (𝐡 ↑m 𝐴))
3 elmapfn 8861 . . . . . . 7 (𝑔 ∈ (𝐡 ↑m 𝐴) β†’ 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔 Fn 𝐴)
5 simplr 766 . . . . . . . . . 10 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ 𝑔 ∈ 𝑋)
6 dffn3 6724 . . . . . . . . . . . 12 (𝑔 Fn 𝐴 ↔ 𝑔:𝐴⟢ran 𝑔)
74, 6sylib 217 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔:𝐴⟢ran 𝑔)
87ffvelcdmda 7080 . . . . . . . . . 10 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ (π‘”β€˜π‘₯) ∈ ran 𝑔)
9 rneq 5929 . . . . . . . . . . . 12 (𝑓 = 𝑔 β†’ ran 𝑓 = ran 𝑔)
109eleq2d 2813 . . . . . . . . . . 11 (𝑓 = 𝑔 β†’ ((π‘”β€˜π‘₯) ∈ ran 𝑓 ↔ (π‘”β€˜π‘₯) ∈ ran 𝑔))
1110rspcev 3606 . . . . . . . . . 10 ((𝑔 ∈ 𝑋 ∧ (π‘”β€˜π‘₯) ∈ ran 𝑔) β†’ βˆƒπ‘“ ∈ 𝑋 (π‘”β€˜π‘₯) ∈ ran 𝑓)
125, 8, 11syl2anc 583 . . . . . . . . 9 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ βˆƒπ‘“ ∈ 𝑋 (π‘”β€˜π‘₯) ∈ ran 𝑓)
13 eliun 4994 . . . . . . . . 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 6142 . . . . . . . 8 ran βˆͺ 𝑋 = βˆͺ 𝑓 ∈ 𝑋 ran 𝑓
1614, 15eleqtrrdi 2838 . . . . . . 7 (((πœ‘ ∧ 𝑔 ∈ 𝑋) ∧ π‘₯ ∈ 𝐴) β†’ (π‘”β€˜π‘₯) ∈ ran βˆͺ 𝑋)
1716ralrimiva 3140 . . . . . 6 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ βˆ€π‘₯ ∈ 𝐴 (π‘”β€˜π‘₯) ∈ ran βˆͺ 𝑋)
184, 17jca 511 . . . . 5 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ (𝑔 Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 (π‘”β€˜π‘₯) ∈ ran βˆͺ 𝑋))
19 ffnfv 7114 . . . . 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 7440 . . . . . . . . 9 (πœ‘ β†’ (𝐡 ↑m 𝐴) ∈ V)
2221, 1ssexd 5317 . . . . . . . 8 (πœ‘ β†’ 𝑋 ∈ V)
2322uniexd 7729 . . . . . . 7 (πœ‘ β†’ βˆͺ 𝑋 ∈ V)
24 rnexg 7892 . . . . . . 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 8836 . . . . 5 (πœ‘ β†’ (𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟢ran βˆͺ 𝑋))
2827adantr 480 . . . 4 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ (𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴) ↔ 𝑔:𝐴⟢ran βˆͺ 𝑋))
2920, 28mpbird 257 . . 3 ((πœ‘ ∧ 𝑔 ∈ 𝑋) β†’ 𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴))
3029ralrimiva 3140 . 2 (πœ‘ β†’ βˆ€π‘” ∈ 𝑋 𝑔 ∈ (ran βˆͺ 𝑋 ↑m 𝐴))
31 dfss3 3965 . 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 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   βŠ† wss 3943  βˆͺ cuni 4902  βˆͺ ciun 4990  ran crn 5670   Fn wfn 6532  βŸΆwf 6533  β€˜cfv 6537  (class class class)co 7405   ↑m cmap 8822
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-map 8824
This theorem is referenced by:  unirnmapsn  44485
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