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Theorem unirnmap 41824
 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 3918 . . . . . . 7 ((𝜑𝑔𝑋) → 𝑔 ∈ (𝐵m 𝐴))
3 elmapfn 8416 . . . . . . 7 (𝑔 ∈ (𝐵m 𝐴) → 𝑔 Fn 𝐴)
42, 3syl 17 . . . . . 6 ((𝜑𝑔𝑋) → 𝑔 Fn 𝐴)
5 simplr 768 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → 𝑔𝑋)
6 dffn3 6503 . . . . . . . . . . . 12 (𝑔 Fn 𝐴𝑔:𝐴⟶ran 𝑔)
74, 6sylib 221 . . . . . . . . . . 11 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑔)
87ffvelrnda 6832 . . . . . . . . . 10 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
9 rneq 5774 . . . . . . . . . . . 12 (𝑓 = 𝑔 → ran 𝑓 = ran 𝑔)
109eleq2d 2878 . . . . . . . . . . 11 (𝑓 = 𝑔 → ((𝑔𝑥) ∈ ran 𝑓 ↔ (𝑔𝑥) ∈ ran 𝑔))
1110rspcev 3574 . . . . . . . . . 10 ((𝑔𝑋 ∧ (𝑔𝑥) ∈ ran 𝑔) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
125, 8, 11syl2anc 587 . . . . . . . . 9 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
13 eliun 4888 . . . . . . . . 9 ((𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓 ↔ ∃𝑓𝑋 (𝑔𝑥) ∈ ran 𝑓)
1412, 13sylibr 237 . . . . . . . 8 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ 𝑓𝑋 ran 𝑓)
15 rnuni 5978 . . . . . . . 8 ran 𝑋 = 𝑓𝑋 ran 𝑓
1614, 15eleqtrrdi 2904 . . . . . . 7 (((𝜑𝑔𝑋) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑋)
1716ralrimiva 3152 . . . . . 6 ((𝜑𝑔𝑋) → ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋)
184, 17jca 515 . . . . 5 ((𝜑𝑔𝑋) → (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
19 ffnfv 6863 . . . . 5 (𝑔:𝐴⟶ran 𝑋 ↔ (𝑔 Fn 𝐴 ∧ ∀𝑥𝐴 (𝑔𝑥) ∈ ran 𝑋))
2018, 19sylibr 237 . . . 4 ((𝜑𝑔𝑋) → 𝑔:𝐴⟶ran 𝑋)
21 ovexd 7174 . . . . . . . . 9 (𝜑 → (𝐵m 𝐴) ∈ V)
2221, 1ssexd 5195 . . . . . . . 8 (𝜑𝑋 ∈ V)
2322uniexd 7452 . . . . . . 7 (𝜑 𝑋 ∈ V)
24 rnexg 7599 . . . . . . 7 ( 𝑋 ∈ V → ran 𝑋 ∈ V)
2523, 24syl 17 . . . . . 6 (𝜑 → ran 𝑋 ∈ V)
26 unirnmap.a . . . . . 6 (𝜑𝐴𝑉)
2725, 26elmapd 8407 . . . . 5 (𝜑 → (𝑔 ∈ (ran 𝑋m 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2827adantr 484 . . . 4 ((𝜑𝑔𝑋) → (𝑔 ∈ (ran 𝑋m 𝐴) ↔ 𝑔:𝐴⟶ran 𝑋))
2920, 28mpbird 260 . . 3 ((𝜑𝑔𝑋) → 𝑔 ∈ (ran 𝑋m 𝐴))
3029ralrimiva 3152 . 2 (𝜑 → ∀𝑔𝑋 𝑔 ∈ (ran 𝑋m 𝐴))
31 dfss3 3906 . 2 (𝑋 ⊆ (ran 𝑋m 𝐴) ↔ ∀𝑔𝑋 𝑔 ∈ (ran 𝑋m 𝐴))
3230, 31sylibr 237 1 (𝜑𝑋 ⊆ (ran 𝑋m 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2112  ∀wral 3109  ∃wrex 3110  Vcvv 3444   ⊆ wss 3884  ∪ cuni 4803  ∪ ciun 4884  ran crn 5524   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328  (class class class)co 7139   ↑m cmap 8393 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-map 8395 This theorem is referenced by:  unirnmapsn  41830
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