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Mirrors > Home > MPE Home > Th. List > axrep6g | Structured version Visualization version GIF version |
Description: axrep6 5250 in class notation. It is equivalent to both ax-rep 5243 and abrexexg 7894, 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 3309 | . . . . . 6 ⊢ (𝑧 = 𝐴 → (∃𝑥 ∈ 𝑧 𝜓 ↔ ∃𝑥 ∈ 𝐴 𝜓)) | |
2 | 1 | abbidv 2802 | . . . . 5 ⊢ (𝑧 = 𝐴 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓}) |
3 | 2 | eleq1d 2819 | . . . 4 ⊢ (𝑧 = 𝐴 → ({𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)) |
4 | 3 | imbi2d 341 | . . 3 ⊢ (𝑧 = 𝐴 → ((∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) ↔ (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V))) |
5 | axrep6 5250 | . . . 4 ⊢ (∀𝑥∃*𝑦𝜓 → ∃𝑤∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓)) | |
6 | abbi 2801 | . . . . . 6 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ 𝑦 ∈ 𝑤} = {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓}) | |
7 | abid2 2872 | . . . . . . 7 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} = 𝑤 | |
8 | vex 3448 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
9 | 7, 8 | eqeltri 2830 | . . . . . 6 ⊢ {𝑦 ∣ 𝑦 ∈ 𝑤} ∈ V |
10 | 6, 9 | eqeltrrdi 2843 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑤 ↔ ∃𝑥 ∈ 𝑧 𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝑧 𝜓} ∈ V) |
11 | 10 | exlimiv 1934 | . . . 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 3524 | . 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥∃*𝑦𝜓 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V)) |
14 | 13 | imp 408 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥∃*𝑦𝜓) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜓} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃*wmo 2533 {cab 2710 ∃wrex 3070 Vcvv 3444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-rep 5243 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-v 3446 |
This theorem is referenced by: funimaexg 6588 abrexexg 7894 |
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