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Mirrors > Home > MPE Home > Th. List > Mathboxes > abrexexd | Structured version Visualization version GIF version |
Description: Existence of a class abstraction of existentially restricted sets. (Contributed by Thierry Arnoux, 10-May-2017.) |
Ref | Expression |
---|---|
abrexexd.0 | ⊢ Ⅎ𝑥𝐴 |
abrexexd.1 | ⊢ (𝜑 → 𝐴 ∈ V) |
Ref | Expression |
---|---|
abrexexd | ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnopab 5820 | . . 3 ⊢ ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | df-mpt 5139 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 2 | rneqi 5801 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | df-rex 3144 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) | |
5 | 4 | abbii 2886 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
6 | 1, 3, 5 | 3eqtr4i 2854 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
7 | abrexexd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
8 | funmpt 6387 | . . . 4 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | eqid 2821 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
10 | 9 | dmmpt 6088 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
11 | abrexexd.0 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
12 | 11 | rabexgfGS 30256 | . . . . 5 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ∈ V) |
13 | 10, 12 | eqeltrid 2917 | . . . 4 ⊢ (𝐴 ∈ V → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
14 | funex 6976 | . . . 4 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
15 | 8, 13, 14 | sylancr 589 | . . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
16 | rnexg 7608 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
17 | 7, 15, 16 | 3syl 18 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
18 | 6, 17 | eqeltrrid 2918 | 1 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 {cab 2799 Ⅎwnfc 2961 ∃wrex 3139 {crab 3142 Vcvv 3494 {copab 5120 ↦ cmpt 5138 dom cdm 5549 ran crn 5550 Fun wfun 6343 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: esumc 31305 |
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