![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 5947 | . . 3 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
2 | df-mpt 5225 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
3 | 2 | rneqi 5930 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ran {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
4 | df-rex 3065 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)) | |
5 | 4 | abbii 2796 | . . 3 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
6 | 1, 3, 5 | 3eqtr4i 2764 | . 2 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} |
7 | abrexexd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ V) | |
8 | funmpt 6580 | . . . 4 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
9 | eqid 2726 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
10 | 9 | dmmpt 6233 | . . . . 5 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
11 | abrexexd.0 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
12 | 11 | rabexgfGS 32248 | . . . . 5 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ∈ V) |
13 | 10, 12 | eqeltrid 2831 | . . . 4 ⊢ (𝐴 ∈ V → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
14 | funex 7216 | . . . 4 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
15 | 8, 13, 14 | sylancr 586 | . . 3 ⊢ (𝐴 ∈ V → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
16 | rnexg 7892 | . . 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 2832 | 1 ⊢ (𝜑 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∃wex 1773 ∈ wcel 2098 {cab 2703 Ⅎwnfc 2877 ∃wrex 3064 {crab 3426 Vcvv 3468 {copab 5203 ↦ cmpt 5224 dom cdm 5669 ran crn 5670 Fun wfun 6531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 |
This theorem is referenced by: esumc 33579 |
Copyright terms: Public domain | W3C validator |