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Mirrors > Home > MPE Home > Th. List > elrab2w | Structured version Visualization version GIF version |
Description: Membership in a restricted class abstraction. This is to elrab2 3588 what elab2gw 3569 is to elab2g 3572. (Contributed by SN, 2-Sep-2024.) |
Ref | Expression |
---|---|
elrab2w.1 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
elrab2w.2 | ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) |
elrab2w.3 | ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑} |
Ref | Expression |
---|---|
elrab2w | ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3415 | . 2 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ V) | |
2 | elex 3415 | . . 3 ⊢ (𝐴 ∈ 𝐵 → 𝐴 ∈ V) | |
3 | 2 | adantr 484 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜒) → 𝐴 ∈ V) |
4 | eleq1w 2815 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
5 | elrab2w.1 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
6 | 4, 5 | anbi12d 634 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ 𝜓))) |
7 | eleq1 2820 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
8 | elrab2w.2 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝜓 ↔ 𝜒)) | |
9 | 7, 8 | anbi12d 634 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝜓) ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))) |
10 | elrab2w.3 | . . . 4 ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ 𝜑} | |
11 | df-rab 3062 | . . . 4 ⊢ {𝑥 ∈ 𝐵 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} | |
12 | 10, 11 | eqtri 2761 | . . 3 ⊢ 𝐶 = {𝑥 ∣ (𝑥 ∈ 𝐵 ∧ 𝜑)} |
13 | 6, 9, 12 | elab2gw 3569 | . 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜒))) |
14 | 1, 3, 13 | pm5.21nii 383 | 1 ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 {cab 2716 {crab 3057 Vcvv 3397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-rab 3062 df-v 3399 |
This theorem is referenced by: elom 7596 |
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