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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtlem11 | Structured version Visualization version GIF version |
Description: Lemma for prter2 36581. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
Ref | Expression |
---|---|
prtlem11 | ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eceq1 8407 | . . . 4 ⊢ (𝑥 = 𝐶 → [𝑥] ∼ = [𝐶] ∼ ) | |
2 | 1 | rspceeqv 3542 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ ) |
3 | elqsg 8428 | . . 3 ⊢ (𝐵 ∈ 𝐷 → (𝐵 ∈ (𝐴 / ∼ ) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ )) | |
4 | 2, 3 | syl5ibr 249 | . 2 ⊢ (𝐵 ∈ 𝐷 → ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 ∈ (𝐴 / ∼ ))) |
5 | 4 | expd 419 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∃wrex 3052 [cec 8367 / cqs 8368 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-br 5040 df-opab 5102 df-xp 5542 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ec 8371 df-qs 8375 |
This theorem is referenced by: prter2 36581 |
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