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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 3069 | . . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | 1 | abbii 2807 | . . 3 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} |
3 | rnopab 5968 | . . 3 ⊢ ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} = {𝑥 ∣ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)} | |
4 | 2, 3 | eqtr4i 2766 | . 2 ⊢ {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} = ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
5 | dmopabss 5932 | . . . . 5 ⊢ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 | |
6 | ssexg 5329 | . . . . 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 6603 | . . . . . . 7 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑦∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
9 | abrexdomjm.1 | . . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑) | |
10 | moanimv 2617 | . . . . . . . 8 ⊢ (∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) ↔ (𝑦 ∈ 𝐴 → ∃*𝑥𝜑)) | |
11 | 9, 10 | mpbir 231 | . . . . . . 7 ⊢ ∃*𝑥(𝑦 ∈ 𝐴 ∧ 𝜑) |
12 | 8, 11 | mpgbir 1796 | . . . . . 6 ⊢ Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} |
13 | 12 | a1i 11 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
14 | funfn 6598 | . . . . 5 ⊢ (Fun {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ↔ {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) | |
15 | 13, 14 | sylib 218 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} Fn dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)}) |
16 | fnrndomg 10574 | . . . 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 9039 | . . . 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 9046 | . . 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 584 | . 2 ⊢ (𝐴 ∈ 𝑉 → ran {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝜑)} ≼ 𝐴) |
22 | 4, 21 | eqbrtrid 5183 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∣ ∃𝑦 ∈ 𝐴 𝜑} ≼ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∃wex 1776 ∈ wcel 2106 ∃*wmo 2536 {cab 2712 ∃wrex 3068 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 {copab 5210 dom cdm 5689 ran crn 5690 Fun wfun 6557 Fn wfn 6558 ≼ cdom 8982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-ac2 10501 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-card 9977 df-acn 9980 df-ac 10154 |
This theorem is referenced by: abrexdom2jm 32536 |
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