Mathbox for Rodolfo Medina |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prtlem11 | Structured version Visualization version GIF version |
Description: Lemma for prter2 33184. (Contributed by Rodolfo Medina, 12-Oct-2010.) |
Ref | Expression |
---|---|
prtlem11 | ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | risset 3044 | . . . 4 ⊢ (𝐶 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐶) | |
2 | r19.41v 3070 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) ↔ (∃𝑥 ∈ 𝐴 𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ )) | |
3 | eceq1 7669 | . . . . . . 7 ⊢ (𝑥 = 𝐶 → [𝑥] ∼ = [𝐶] ∼ ) | |
4 | eqtr3 2631 | . . . . . . . 8 ⊢ (([𝑥] ∼ = [𝐶] ∼ ∧ 𝐵 = [𝐶] ∼ ) → [𝑥] ∼ = 𝐵) | |
5 | 4 | eqcomd 2616 | . . . . . . 7 ⊢ (([𝑥] ∼ = [𝐶] ∼ ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 = [𝑥] ∼ ) |
6 | 3, 5 | sylan 487 | . . . . . 6 ⊢ ((𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 = [𝑥] ∼ ) |
7 | 6 | reximi 2994 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 (𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ ) |
8 | 2, 7 | sylbir 224 | . . . 4 ⊢ ((∃𝑥 ∈ 𝐴 𝑥 = 𝐶 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ ) |
9 | 1, 8 | sylanb 488 | . . 3 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ ) |
10 | elqsg 7685 | . . 3 ⊢ (𝐵 ∈ 𝐷 → (𝐵 ∈ (𝐴 / ∼ ) ↔ ∃𝑥 ∈ 𝐴 𝐵 = [𝑥] ∼ )) | |
11 | 9, 10 | syl5ibr 235 | . 2 ⊢ (𝐵 ∈ 𝐷 → ((𝐶 ∈ 𝐴 ∧ 𝐵 = [𝐶] ∼ ) → 𝐵 ∈ (𝐴 / ∼ ))) |
12 | 11 | expd 451 | 1 ⊢ (𝐵 ∈ 𝐷 → (𝐶 ∈ 𝐴 → (𝐵 = [𝐶] ∼ → 𝐵 ∈ (𝐴 / ∼ )))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃wrex 2897 [cec 7627 / cqs 7628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-cnv 5046 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-ec 7631 df-qs 7635 |
This theorem is referenced by: prter2 33184 |
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