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Theorem abrexdom 34874
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 3148 . . . 4 (∃𝑦𝐴 𝜑 ↔ ∃𝑦(𝑦𝐴𝜑))
21abbii 2890 . . 3 {𝑥 ∣ ∃𝑦𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦𝐴𝜑)}
3 rnopab 5824 . . 3 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑦𝐴𝜑)}
42, 3eqtr4i 2851 . 2 {𝑥 ∣ ∃𝑦𝐴 𝜑} = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)}
5 dmopabss 5785 . . . . 5 dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴
6 ssexg 5223 . . . . 5 ((dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴𝐴𝑉) → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∈ V)
75, 6mpan 686 . . . 4 (𝐴𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∈ V)
8 funopab 6386 . . . . . . 7 (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ↔ ∀𝑦∃*𝑥(𝑦𝐴𝜑))
9 abrexdom.1 . . . . . . . 8 (𝑦𝐴 → ∃*𝑥𝜑)
10 moanimv 2701 . . . . . . . 8 (∃*𝑥(𝑦𝐴𝜑) ↔ (𝑦𝐴 → ∃*𝑥𝜑))
119, 10mpbir 232 . . . . . . 7 ∃*𝑥(𝑦𝐴𝜑)
128, 11mpgbir 1793 . . . . . 6 Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)}
1312a1i 11 . . . . 5 (𝐴𝑉 → Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
14 funfn 6381 . . . . 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 9950 . . . 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 8547 . . . 4 (𝐴𝑉 → (dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴))
195, 18mpi 20 . . 3 (𝐴𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴)
20 domtr 8554 . . 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 5097 1 (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wex 1773  wcel 2106  ∃*wmo 2613  {cab 2802  wrex 3143  Vcvv 3499  wss 3939   class class class wbr 5062  {copab 5124  dom cdm 5553  ran crn 5554  Fun wfun 6345   Fn wfn 6346  cdom 8499
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 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-ac2 9877
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-se 5513  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-isom 6360  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-card 9360  df-acn 9363  df-ac 9534
This theorem is referenced by:  abrexdom2  34875
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