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Mirrors > Home > MPE Home > Th. List > rusgrnumwwlkslem | Structured version Visualization version GIF version |
Description: Lemma for rusgrnumwwlks 29495. (Contributed by Alexander van der Vekens, 23-Aug-2018.) |
Ref | Expression |
---|---|
rusgrnumwwlkslem | ⊢ (𝑌 ∈ {𝑤 ∈ 𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ 𝜓)} = {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq1 6889 | . . . 4 ⊢ (𝑤 = 𝑌 → (𝑤‘0) = (𝑌‘0)) | |
2 | 1 | eqeq1d 2732 | . . 3 ⊢ (𝑤 = 𝑌 → ((𝑤‘0) = 𝑃 ↔ (𝑌‘0) = 𝑃)) |
3 | 2 | elrab 3682 | . 2 ⊢ (𝑌 ∈ {𝑤 ∈ 𝑍 ∣ (𝑤‘0) = 𝑃} ↔ (𝑌 ∈ 𝑍 ∧ (𝑌‘0) = 𝑃)) |
4 | ibar 527 | . . . . 5 ⊢ ((𝑌‘0) = 𝑃 → ((𝜑 ∧ 𝜓) ↔ ((𝑌‘0) = 𝑃 ∧ (𝜑 ∧ 𝜓)))) | |
5 | 3anass 1093 | . . . . . 6 ⊢ (((𝑌‘0) = 𝑃 ∧ 𝜑 ∧ 𝜓) ↔ ((𝑌‘0) = 𝑃 ∧ (𝜑 ∧ 𝜓))) | |
6 | 3ancoma 1096 | . . . . . 6 ⊢ (((𝑌‘0) = 𝑃 ∧ 𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)) | |
7 | 5, 6 | bitr3i 276 | . . . . 5 ⊢ (((𝑌‘0) = 𝑃 ∧ (𝜑 ∧ 𝜓)) ↔ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)) |
8 | 4, 7 | bitrdi 286 | . . . 4 ⊢ ((𝑌‘0) = 𝑃 → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓))) |
9 | 8 | ad2antlr 723 | . . 3 ⊢ (((𝑌 ∈ 𝑍 ∧ (𝑌‘0) = 𝑃) ∧ 𝑤 ∈ 𝑋) → ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓))) |
10 | 9 | rabbidva 3437 | . 2 ⊢ ((𝑌 ∈ 𝑍 ∧ (𝑌‘0) = 𝑃) → {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ 𝜓)} = {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)}) |
11 | 3, 10 | sylbi 216 | 1 ⊢ (𝑌 ∈ {𝑤 ∈ 𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ 𝜓)} = {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 {crab 3430 ‘cfv 6542 0cc0 11112 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-3an 1087 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-rab 3431 df-v 3474 df-in 3954 df-ss 3964 df-uni 4908 df-br 5148 df-iota 6494 df-fv 6550 |
This theorem is referenced by: rusgrnumwwlks 29495 |
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