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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 2918 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | abbii 2739 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} |
3 | rnopab 5370 | . . 3 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} | |
4 | 2, 3 | eqtr4i 2647 | . 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
5 | dmopabss 5336 | . . . . 5 ⊢ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
6 | ssexg 4804 | . . . . 5 ⊢ ((dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V) | |
7 | 5, 6 | mpan 706 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ∈ V) |
8 | funopab 5923 | . . . . . . 7 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
9 | abrexdomjm.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) | |
10 | moanimv 2531 | . . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)) | |
11 | 9, 10 | mpbir 221 | . . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
12 | 8, 11 | mpgbir 1726 | . . . . . 6 ⊢ Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
14 | funfn 5918 | . . . . 5 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) | |
15 | 13, 14 | sylib 208 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
16 | fnrndomg 9358 | . . . 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 8001 | . . . 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 8009 | . . 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 693 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴) |
22 | 4, 21 | syl5eqbr 4688 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∃wex 1704 ∈ wcel 1990 ∃*wmo 2471 {cab 2608 ∃wrex 2913 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 {copab 4712 dom cdm 5114 ran crn 5115 Fun wfun 5882 Fn wfn 5883 ≼ cdom 7953 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-ac2 9285 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-card 8765 df-acn 8768 df-ac 8939 |
This theorem is referenced by: abrexdom2jm 29346 |
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