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Theorem abrexdom 34886
Description: An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
abrexdom.1 (𝑦𝐴 → ∃*𝑥𝜑)
Assertion
Ref Expression
abrexdom (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem abrexdom
StepHypRef Expression
1 df-rex 3141 . . . 4 (∃𝑦𝐴 𝜑 ↔ ∃𝑦(𝑦𝐴𝜑))
21abbii 2883 . . 3 {𝑥 ∣ ∃𝑦𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦𝐴𝜑)}
3 rnopab 5819 . . 3 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑦𝐴𝜑)}
42, 3eqtr4i 2844 . 2 {𝑥 ∣ ∃𝑦𝐴 𝜑} = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)}
5 dmopabss 5780 . . . . 5 dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴
6 ssexg 5218 . . . . 5 ((dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴𝐴𝑉) → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∈ V)
75, 6mpan 686 . . . 4 (𝐴𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∈ V)
8 funopab 6383 . . . . . . 7 (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ↔ ∀𝑦∃*𝑥(𝑦𝐴𝜑))
9 abrexdom.1 . . . . . . . 8 (𝑦𝐴 → ∃*𝑥𝜑)
10 moanimv 2697 . . . . . . . 8 (∃*𝑥(𝑦𝐴𝜑) ↔ (𝑦𝐴 → ∃*𝑥𝜑))
119, 10mpbir 232 . . . . . . 7 ∃*𝑥(𝑦𝐴𝜑)
128, 11mpgbir 1791 . . . . . 6 Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)}
1312a1i 11 . . . . 5 (𝐴𝑉 → Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
14 funfn 6378 . . . . 5 (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ↔ {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
1513, 14sylib 219 . . . 4 (𝐴𝑉 → {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
16 fnrndomg 9946 . . . 4 (dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∈ V → ({⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)}))
177, 15, 16sylc 65 . . 3 (𝐴𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
18 ssdomg 8543 . . . 4 (𝐴𝑉 → (dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴))
195, 18mpi 20 . . 3 (𝐴𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴)
20 domtr 8550 . . 3 ((ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∧ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴) → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴)
2117, 19, 20syl2anc 584 . 2 (𝐴𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴)
224, 21eqbrtrid 5092 1 (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1771  wcel 2105  ∃*wmo 2613  {cab 2796  wrex 3136  Vcvv 3492  wss 3933   class class class wbr 5057  {copab 5119  dom cdm 5548  ran crn 5549  Fun wfun 6342   Fn wfn 6343  cdom 8495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-card 9356  df-acn 9359  df-ac 9530
This theorem is referenced by:  abrexdom2  34887
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