Theorem abrexdom 38226

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

Hypothesis

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

Assertion

Ref	Expression
abrexdom	⊢ (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 abrexdom

Step	Hyp	Ref	Expression
1		df-rex 3087	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑))
2	1	abbii 2829	. . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)}
3		rnopab 5930	. . 3 ⊢ ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)}
4	2, 3	eqtr4i 2788	. 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
5		dmopabss 5894	. . . . 5 ⊢ dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴
6		ssexg 5279	. . . . 5 ⊢ ((dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V)
7	5, 6	mpan 700	. . . 4 ⊢ (𝐴 ∈ 𝑉 → dom {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V)
8		funopab 6556	. . . . . . 7 ⊢ (Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑))
9		abrexdom.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)
10		moanimv 2646	. . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑))
11	9, 10	mpbir 233	. . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)
12	8, 11	mpgbir 1819	. . . . . 6 ⊢ Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}
13	12	a1i 11	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)})
14		funfn 6551	. . . . 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 10493	. . . 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 8981	. . . 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 8988	. . 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 593	. 2 ⊢ (𝐴 ∈ 𝑉 → ran {⟨𝑦, 𝑥⟩ ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴)
22	4, 21	eqbrtrid 5135	1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴)

Syntax hints:   → wi 4   ∧ wa 399  ∃wex 1799   ∈ wcel 2142  ∃*wmo 2564  {cab 2740  ∃wrex 3086  Vcvv 3454   ⊆ wss 3904   class class class wbr 5100  {copab 5162  dom cdm 5647  ran crn 5648  Fun wfun 6515   Fn wfn 6516   ≼ cdom 8925

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-ac2 10420

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-card 9897  df-acn 9900  df-ac 10072

This theorem is referenced by:  abrexdom2  38227