Theorem abrexdomjm 32535

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 3077	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2	1	abbii 2812	. . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)}
3		rnopab 5979	. . 3 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)}
4	2, 3	eqtr4i 2771	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
5		dmopabss 5943	. . . . 5 ⊢ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6		ssexg 5341	. . . . 5 ⊢ ((dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V)
7	5, 6	mpan 689	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V)
8		funopab 6613	. . . . . . 7 ⊢ (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
9		abrexdomjm.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)
10		moanimv 2622	. . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑))
11	9, 10	mpbir 231	. . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
12	8, 11	mpgbir 1797	. . . . . 6 ⊢ Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
13	12	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
14		funfn 6608	. . . . 5 ⊢ (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
15	13, 14	sylib 218	. . . 4 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
16		fnrndomg 10605	. . . 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 9060	. . . 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 9067	. . 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 583	. 2 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴)
22	4, 21	eqbrtrid 5201	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1777   ∈ wcel 2108  ∃*wmo 2541  {cab 2717  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976   class class class wbr 5166  {copab 5228  dom cdm 5700  ran crn 5701  Fun wfun 6567   Fn wfn 6568   ≼ cdom 9001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-card 10008  df-acn 10011  df-ac 10185

This theorem is referenced by:  abrexdom2jm  32536