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Mirrors > Home > MPE Home > Th. List > axrep6g | Structured version Visualization version GIF version |
Description: axrep6 5287 in class notation. It is equivalent to both ax-rep 5280 and abrexexg 7959, providing a direct link between the two. (Contributed by SN, 11-Dec-2024.) |
Ref | Expression |
---|---|
axrep6g | ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 3317 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | 1 | abbidv 2797 | . . . . 5 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓}) |
3 | 2 | eleq1d 2814 | . . . 4 ⊢ (𝑧 = 𝐴 → ({𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)) |
4 | 3 | imbi2d 340 | . . 3 ⊢ (𝑧 = 𝐴 → ((∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) ↔ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V))) |
5 | axrep6 5287 | . . . 4 ⊢ (∀𝑥∃*𝑦𝜓 → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓)) | |
6 | abbi 2796 | . . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ 𝑦 ∈ 𝑤} = {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓}) | |
7 | abid2 2867 | . . . . . . 7 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} = 𝑤 | |
8 | vex 3474 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
9 | 7, 8 | eqeltri 2825 | . . . . . 6 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} ∈ V |
10 | 6, 9 | eqeltrrdi 2838 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) |
11 | 10 | exlimiv 1926 | . . . 4 ⊢ (∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) |
12 | 5, 11 | syl 17 | . . 3 ⊢ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) |
13 | 4, 12 | vtoclg 3539 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)) |
14 | 13 | imp 406 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ∃*wmo 2528 {cab 2705 ∃wrex 3066 Vcvv 3470 |
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-ext 2699 ax-rep 5280 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-mo 2530 df-clab 2706 df-cleq 2720 df-clel 2806 df-rex 3067 df-v 3472 |
This theorem is referenced by: funimaexg 6634 abrexexg 7959 |
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