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Theorem unirnmap 38871
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 3583 . . . . . . 7 ((𝜑𝑔𝑋) → 𝑔 ∈ (𝐵𝑚 𝐴))
3 elmapfn 7824 . . . . . . 7 (𝑔 ∈ (𝐵𝑚 𝐴) → 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑔𝑋) → 𝑔 Fn 𝐴)
5 simplr 791 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → 𝑔𝑋)
6 dffn3 6011 . . . . . . . . . . . 12 (𝑔 Fn 𝐴𝑔:𝐴⟶ran 𝑔)
74, 6sylib 208 . . . . . . . . . . 11 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑔)
87ffvelrnda 6315 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
9 rneq 5311 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
109eleq2d 2684 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑔𝑥) ∈ ran 𝑓 ↔ (𝑔𝑥) ∈ ran 𝑔))
1110rspcev 3295 . . . . . . . . . 10 ((𝑔𝑋 ∧ (𝑔𝑥) ∈ ran 𝑔) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
125, 8, 11syl2anc 692 . . . . . . . . 9 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
13 eliun 4490 . . . . . . . . 9 ((𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
1412, 13sylibr 224 . . . . . . . 8 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓)
15 rnuni 5503 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
1614, 15syl6eleqr 2709 . . . . . . 7 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑋)
1716ralrimiva 2960 . . . . . 6 ((𝜑𝑔𝑋) → ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋)
184, 17jca 554 . . . . 5 ((𝜑𝑔𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
19 ffnfv 6343 . . . . 5 (𝑔:𝐴⟶ran 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
2018, 19sylibr 224 . . . 4 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑋)
21 ovex 6632 . . . . . . . . . 10 (𝐵𝑚 𝐴) ∈ V
2221a1i 11 . . . . . . . . 9 (𝜑 → (𝐵𝑚 𝐴) ∈ V)
2322, 1ssexd 4765 . . . . . . . 8 (𝜑𝑋 ∈ V)
24 uniexg 6908 . . . . . . . 8 (𝑋 ∈ V → 𝑋 ∈ V)
2523, 24syl 17 . . . . . . 7 (𝜑 𝑋 ∈ V)
26 rnexg 7045 . . . . . . 7 ( 𝑋 ∈ V → ran 𝑋 ∈ V)
2725, 26syl 17 . . . . . 6 (𝜑 → ran 𝑋 ∈ V)
28 unirnmap.a . . . . . 6 (𝜑𝐴𝑉)
2927, 28elmapd 7816 . . . . 5 (𝜑 → (𝑔 ∈ (ran 𝑋𝑚 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
3029adantr 481 . . . 4 ((𝜑𝑔𝑋) → (𝑔 ∈ (ran 𝑋𝑚 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
3120, 30mpbird 247 . . 3 ((𝜑𝑔𝑋) → 𝑔 ∈ (ran 𝑋𝑚 𝐴))
3231ralrimiva 2960 . 2 (𝜑 → ∀𝑔𝑋 𝑔 ∈ (ran 𝑋𝑚 𝐴))
33 dfss3 3573 . 2 (𝑋 ⊆ (ran 𝑋𝑚 𝐴) ↔ ∀𝑔𝑋 𝑔 ∈ (ran 𝑋𝑚 𝐴))
3432, 33sylibr 224 1 (𝜑𝑋 ⊆ (ran 𝑋𝑚 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wss 3555   cuni 4402   ciun 4485  ran crn 5075   Fn wfn 5842  wf 5843  cfv 5847  (class class class)co 6604  𝑚 cmap 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-map 7804
This theorem is referenced by:  unirnmapsn  38877
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