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Mirrors > Home > MPE Home > Th. List > qseq1 | Structured version Visualization version GIF version |
Description: Equality theorem for quotient set. (Contributed by NM, 23-Jul-1995.) |
Ref | Expression |
---|---|
qseq1 | ⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 3310 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶)) | |
2 | 1 | abbidv 2794 | . 2 ⊢ (𝐴 = 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶}) |
3 | df-qs 8731 | . 2 ⊢ (𝐴 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = [𝑥]𝐶} | |
4 | df-qs 8731 | . 2 ⊢ (𝐵 / 𝐶) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑦 = [𝑥]𝐶} | |
5 | 2, 3, 4 | 3eqtr4g 2790 | 1 ⊢ (𝐴 = 𝐵 → (𝐴 / 𝐶) = (𝐵 / 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 {cab 2702 ∃wrex 3059 [cec 8723 / cqs 8724 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-rex 3060 df-qs 8731 |
This theorem is referenced by: qseq1d 8783 qseq12 8785 pi1bas 25009 pstmval 33624 qseq1i 37889 |
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