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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 3061 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | abbii 2796 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} |
3 | rnopab 5960 | . . 3 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} | |
4 | 2, 3 | eqtr4i 2757 | . 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
5 | dmopabss 5925 | . . . . 5 ⊢ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
6 | ssexg 5328 | . . . . 5 ⊢ ((dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V) | |
7 | 5, 6 | mpan 688 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V) |
8 | funopab 6594 | . . . . . . 7 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
9 | abrexdom.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) | |
10 | moanimv 2608 | . . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)) | |
11 | 9, 10 | mpbir 230 | . . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
12 | 8, 11 | mpgbir 1794 | . . . . . 6 ⊢ Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
14 | funfn 6589 | . . . . 5 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) | |
15 | 13, 14 | sylib 217 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
16 | fnrndomg 10579 | . . . 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 9031 | . . . 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 9038 | . . 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 582 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴) |
22 | 4, 21 | eqbrtrid 5188 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∃wex 1774 ∈ wcel 2099 ∃*wmo 2527 {cab 2703 ∃wrex 3060 Vcvv 3462 ⊆ wss 3947 class class class wbr 5153 {copab 5215 dom cdm 5682 ran crn 5683 Fun wfun 6548 Fn wfn 6549 ≼ cdom 8972 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-ac2 10506 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-card 9982 df-acn 9985 df-ac 10159 |
This theorem is referenced by: abrexdom2 37432 |
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