Theorem abrexdomjm 30261

Description: An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypothesis

Ref	Expression
abrexdomjm.1	⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)

Assertion

Ref	Expression
abrexdomjm	⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴)

Distinct variable group:   𝑥,𝐴,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem abrexdomjm

Step	Hyp	Ref	Expression
1		df-rex 3144	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2	1	abbii 2886	. . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)}
3		rnopab 5820	. . 3 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)}
4	2, 3	eqtr4i 2847	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
5		dmopabss 5781	. . . . 5 ⊢ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6		ssexg 5219	. . . . 5 ⊢ ((dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V)
7	5, 6	mpan 688	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V)
8		funopab 6384	. . . . . . 7 ⊢ (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
9		abrexdomjm.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)
10		moanimv 2700	. . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑))
11	9, 10	mpbir 233	. . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
12	8, 11	mpgbir 1796	. . . . . 6 ⊢ Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
13	12	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
14		funfn 6379	. . . . 5 ⊢ (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
15	13, 14	sylib 220	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
16		fnrndomg 9952	. . . 4 ⊢ (dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V → ({⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}))
17	7, 15, 16	sylc 65	. . 3 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
18		ssdomg 8549	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴))
19	5, 18	mpi 20	. . 3 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴)
20		domtr 8556	. . 3 ⊢ ((ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∧ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴) → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴)
21	17, 19, 20	syl2anc 586	. 2 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴)
22	4, 21	eqbrtrid 5093	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398  ∃wex 1776   ∈ wcel 2110  ∃*wmo 2616  {cab 2799  ∃wrex 3139  Vcvv 3494   ⊆ wss 3935   class class class wbr 5058  {copab 5120  dom cdm 5549  ran crn 5550  Fun wfun 6343   Fn wfn 6344   ≼ cdom 8501

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-ac2 9879

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-er 8283  df-map 8402  df-en 8504  df-dom 8505  df-card 9362  df-acn 9365  df-ac 9536

This theorem is referenced by:  abrexdom2jm  30262