Theorem abrexdomjm 32443

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 3055	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2	1	abbii 2797	. . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)}
3		rnopab 5921	. . 3 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)}
4	2, 3	eqtr4i 2756	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
5		dmopabss 5885	. . . . 5 ⊢ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6		ssexg 5281	. . . . 5 ⊢ ((dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V)
7	5, 6	mpan 690	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V)
8		funopab 6554	. . . . . . 7 ⊢ (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
9		abrexdomjm.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)
10		moanimv 2613	. . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑))
11	9, 10	mpbir 231	. . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
12	8, 11	mpgbir 1799	. . . . . 6 ⊢ Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
13	12	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
14		funfn 6549	. . . . 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 10496	. . . 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 8974	. . . 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 8981	. . 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 584	. 2 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴)
22	4, 21	eqbrtrid 5145	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2532  {cab 2708  ∃wrex 3054  Vcvv 3450   ⊆ wss 3917   class class class wbr 5110  {copab 5172  dom cdm 5641  ran crn 5642  Fun wfun 6508   Fn wfn 6509   ≼ cdom 8919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-ac2 10423

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-card 9899  df-acn 9902  df-ac 10076

This theorem is referenced by:  abrexdom2jm  32444