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 2921 | . . 3 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} | |
2 | 1 | sseq1i 3629 | . 2 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵) |
3 | rabssf.1 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
4 | 3 | abssf 39295 | . 2 ⊢ ({𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐵 ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵)) |
5 | impexp 462 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
6 | 5 | albii 1747 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) |
7 | df-ral 2917 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝑥 ∈ 𝐵))) | |
8 | 6, 7 | bitr4i 267 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐵) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
9 | 2, 4, 8 | 3bitri 286 | 1 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 {cab 2608 Ⅎwnfc 2751 ∀wral 2912 {crab 2916 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-in 3581 df-ss 3588 |
This theorem is referenced by: rabssd 39332 supminfxr2 39699 preimageiingt 40930 preimaleiinlt 40931 smfmullem4 41001 |
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