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Mirrors > Home > MPE Home > Th. List > Mathboxes > abrexdomjm | 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 |
---|---|
abrexdomjm.1 | ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) |
Ref | Expression |
---|---|
abrexdomjm | ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3060 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | abbii 2804 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} |
3 | rnopab 5812 | . . 3 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} | |
4 | 2, 3 | eqtr4i 2765 | . 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
5 | dmopabss 5776 | . . . . 5 ⊢ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
6 | ssexg 5205 | . . . . 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 6404 | . . . . . . 7 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
9 | abrexdomjm.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) | |
10 | moanimv 2618 | . . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)) | |
11 | 9, 10 | mpbir 234 | . . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
12 | 8, 11 | mpgbir 1807 | . . . . . 6 ⊢ Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
14 | funfn 6399 | . . . . 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 10133 | . . . 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 8663 | . . . 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 8670 | . . 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 5078 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∃wex 1787 ∈ wcel 2110 ∃*wmo 2535 {cab 2712 ∃wrex 3055 Vcvv 3401 ⊆ wss 3857 class class class wbr 5043 {copab 5105 dom cdm 5540 ran crn 5541 Fun wfun 6363 Fn wfn 6364 ≼ cdom 8613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-ac2 10060 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-card 9538 df-acn 9541 df-ac 9713 |
This theorem is referenced by: abrexdom2jm 30544 |
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