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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rabssf | Structured version Visualization version GIF version |
Description: Restricted class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
rabssf.1 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
rabssf | ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3407 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | sseq1i 3976 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵) |
3 | rabssf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 3 | abssf 43414 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵)) |
5 | impexp 452 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
6 | 5 | albii 1822 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) |
7 | df-ral 3062 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
8 | 6, 7 | bitr4i 278 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
9 | 2, 4, 8 | 3bitri 297 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∀wal 1540 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 ∀wral 3061 {crab 3406 ⊆ wss 3914 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rab 3407 df-v 3449 df-in 3921 df-ss 3931 |
This theorem is referenced by: rabssd 43444 supminfxr2 43794 preimageiingt 45051 preimaleiinlt 45052 smfmullem4 45125 |
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