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Mirrors > Home > MPE Home > Th. List > Mathboxes > ssrabf | Structured version Visualization version GIF version |
Description: Subclass of a restricted class abstraction. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
ssrabf.1 | ⊢ Ⅎ𝑥𝐵 |
ssrabf.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
ssrabf | ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rab 3224 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | sseq2i 3952 | . 2 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ 𝐵 ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)}) |
3 | ssrabf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 3 | ssabf 42674 | . 2 ⊢ (𝐵 ⊆ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜑))) |
5 | ssrabf.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
6 | 3, 5 | dfss3f 3914 | . . . 4 ⊢ (𝐵 ⊆ 𝐴 ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐴) |
7 | 6 | anbi1i 623 | . . 3 ⊢ ((𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑) ↔ (∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
8 | r19.26 3108 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (∀𝑥 ∈ 𝐵 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) | |
9 | df-ral 3060 | . . 3 ⊢ (∀𝑥 ∈ 𝐵 (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∀𝑥(𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜑))) | |
10 | 7, 8, 9 | 3bitr2ri 299 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐵 → (𝑥 ∈ 𝐴 ∧ 𝜑)) ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
11 | 2, 4, 10 | 3bitri 296 | 1 ⊢ (𝐵 ⊆ {𝑥 ∈ 𝐴 ∣ 𝜑} ↔ (𝐵 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1535 ∈ wcel 2101 {cab 2710 Ⅎwnfc 2882 ∀wral 3059 {crab 3221 ⊆ wss 3889 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-10 2132 ax-11 2149 ax-12 2166 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-tru 1540 df-ex 1778 df-nf 1782 df-sb 2063 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2884 df-ral 3060 df-rab 3224 df-v 3436 df-in 3896 df-ss 3906 |
This theorem is referenced by: supminfxr2 43043 pimgtmnf2 44288 smfmullem4 44368 smflimsuplem7 44399 |
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