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Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec2lem1 | Structured version Visualization version GIF version |
Description: Lemma for setrec2 42241. The functional part of 𝐹 has the same values as 𝐹. (Contributed by Emmett Weisz, 4-Mar-2021.) (New usage is discouraged.) |
Ref | Expression |
---|---|
setrec2lem1 | ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvres 6117 | . 2 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)) | |
2 | dmres 5339 | . . . . . . 7 ⊢ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) = ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹) | |
3 | inss1 3795 | . . . . . . 7 ⊢ ({𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ∩ dom 𝐹) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} | |
4 | 2, 3 | eqsstri 3598 | . . . . . 6 ⊢ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) ⊆ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} |
5 | 4 | sseli 3564 | . . . . 5 ⊢ (𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) |
6 | 5 | con3i 149 | . . . 4 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ 𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})) |
7 | ndmfv 6128 | . . . 4 ⊢ (¬ 𝑎 ∈ dom (𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦}) → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅) |
9 | vex 3176 | . . . . . . 7 ⊢ 𝑎 ∈ V | |
10 | breq1 4586 | . . . . . . . 8 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑦 ↔ 𝑎𝐹𝑦)) | |
11 | 10 | eubidv 2478 | . . . . . . 7 ⊢ (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦)) |
12 | 9, 11 | elab 3319 | . . . . . 6 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦) |
13 | 12 | notbii 309 | . . . . 5 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ¬ ∃!𝑦 𝑎𝐹𝑦) |
14 | 13 | biimpi 205 | . . . 4 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ¬ ∃!𝑦 𝑎𝐹𝑦) |
15 | tz6.12-2 6094 | . . . 4 ⊢ (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹‘𝑎) = ∅) | |
16 | 14, 15 | syl 17 | . . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹‘𝑎) = ∅) |
17 | 8, 16 | eqtr4d 2647 | . 2 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)) |
18 | 1, 17 | pm2.61i 175 | 1 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1475 ∈ wcel 1977 ∃!weu 2458 {cab 2596 ∩ cin 3539 ∅c0 3874 class class class wbr 4583 dom cdm 5038 ↾ cres 5040 ‘cfv 5804 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
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-eu 2462 df-mo 2463 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-uni 4373 df-br 4584 df-opab 4644 df-xp 5044 df-dm 5048 df-res 5050 df-iota 5768 df-fv 5812 |
This theorem is referenced by: setrec2 42241 |
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