Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abrexdomjm Structured version   Visualization version   GIF version

Theorem abrexdomjm 28517
Description: An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
abrexdomjm.1 (𝑦𝐴 → ∃*𝑥𝜑)
Assertion
Ref Expression
abrexdomjm (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)
Distinct variable group:   𝑥,𝐴,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem abrexdomjm
StepHypRef Expression
1 df-rex 2806 . . . 4 (∃𝑦𝐴 𝜑 ↔ ∃𝑦(𝑦𝐴𝜑))
21abbii 2630 . . 3 {𝑥 ∣ ∃𝑦𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦𝐴𝜑)}
3 rnopab 5182 . . 3 ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} = {𝑥 ∣ ∃𝑦(𝑦𝐴𝜑)}
42, 3eqtr4i 2539 . 2 {𝑥 ∣ ∃𝑦𝐴 𝜑} = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)}
5 dmopabss 5149 . . . . 5 dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴
6 ssexg 4631 . . . . 5 ((dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴𝐴𝑉) → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∈ V)
75, 6mpan 701 . . . 4 (𝐴𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∈ V)
8 funopab 5722 . . . . . . 7 (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ↔ ∀𝑦∃*𝑥(𝑦𝐴𝜑))
9 abrexdomjm.1 . . . . . . . 8 (𝑦𝐴 → ∃*𝑥𝜑)
10 moanimv 2423 . . . . . . . 8 (∃*𝑥(𝑦𝐴𝜑) ↔ (𝑦𝐴 → ∃*𝑥𝜑))
119, 10mpbir 219 . . . . . . 7 ∃*𝑥(𝑦𝐴𝜑)
128, 11mpgbir 1704 . . . . . 6 Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)}
1312a1i 11 . . . . 5 (𝐴𝑉 → Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
14 funfn 5718 . . . . 5 (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ↔ {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
1513, 14sylib 206 . . . 4 (𝐴𝑉 → {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
16 fnrndomg 9115 . . . 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 62 . . 3 (𝐴𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)})
18 ssdomg 7763 . . . 4 (𝐴𝑉 → (dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ⊆ 𝐴 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴))
195, 18mpi 20 . . 3 (𝐴𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴)
20 domtr 7771 . . 3 ((ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ∧ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴) → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴)
2117, 19, 20syl2anc 690 . 2 (𝐴𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦𝐴𝜑)} ≼ 𝐴)
224, 21syl5eqbr 4516 1 (𝐴𝑉 → {𝑥 ∣ ∃𝑦𝐴 𝜑} ≼ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wex 1694  wcel 1938  ∃*wmo 2363  {cab 2500  wrex 2801  Vcvv 3077  wss 3444   class class class wbr 4481  {copab 4540  dom cdm 4932  ran crn 4933  Fun wfun 5683   Fn wfn 5684  cdom 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-ac2 9044
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-se 4892  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-isom 5698  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-1st 6934  df-2nd 6935  df-wrecs 7169  df-recs 7231  df-er 7505  df-map 7622  df-en 7718  df-dom 7719  df-card 8524  df-acn 8527  df-ac 8698
This theorem is referenced by:  abrexdom2jm  28518
  Copyright terms: Public domain W3C validator