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Mirrors > Home > MPE Home > Th. List > Mathboxes > abrexdom | Structured version Visualization version GIF version |
Description: An indexed set is dominated by the indexing set. (Contributed by Jeff Madsen, 2-Sep-2009.) |
Ref | Expression |
---|---|
abrexdom.1 | ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) |
Ref | Expression |
---|---|
abrexdom | ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3059 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | abbii 2803 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} |
3 | rnopab 5791 | . . 3 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} | |
4 | 2, 3 | eqtr4i 2764 | . 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
5 | dmopabss 5755 | . . . . 5 ⊢ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
6 | ssexg 5188 | . . . . 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 6368 | . . . . . . 7 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
9 | abrexdom.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) | |
10 | moanimv 2622 | . . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)) | |
11 | 9, 10 | mpbir 234 | . . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
12 | 8, 11 | mpgbir 1806 | . . . . . 6 ⊢ Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
14 | funfn 6363 | . . . . 5 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) | |
15 | 13, 14 | sylib 221 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
16 | fnrndomg 10029 | . . . 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 8594 | . . . 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 8601 | . . 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 587 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴) |
22 | 4, 21 | eqbrtrid 5062 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1786 ∈ wcel 2113 ∃*wmo 2538 {cab 2716 ∃wrex 3054 Vcvv 3397 ⊆ wss 3841 class class class wbr 5027 {copab 5089 dom cdm 5519 ran crn 5520 Fun wfun 6327 Fn wfn 6328 ≼ cdom 8546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-ac2 9956 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-card 9434 df-acn 9437 df-ac 9609 |
This theorem is referenced by: abrexdom2 35501 |
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