![]() |
Mathbox for Emmett Weisz |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > setrec2lem1 | Structured version Visualization version GIF version |
Description: Lemma for setrec2 48039. 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 6910 | . 2 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)) | |
2 | nfvres 6932 | . . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = ∅) | |
3 | vex 3473 | . . . . 5 ⊢ 𝑎 ∈ V | |
4 | breq1 5145 | . . . . . 6 ⊢ (𝑥 = 𝑎 → (𝑥𝐹𝑦 ↔ 𝑎𝐹𝑦)) | |
5 | 4 | eubidv 2575 | . . . . 5 ⊢ (𝑥 = 𝑎 → (∃!𝑦 𝑥𝐹𝑦 ↔ ∃!𝑦 𝑎𝐹𝑦)) |
6 | 3, 5 | elab 3665 | . . . 4 ⊢ (𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} ↔ ∃!𝑦 𝑎𝐹𝑦) |
7 | tz6.12-2 6879 | . . . 4 ⊢ (¬ ∃!𝑦 𝑎𝐹𝑦 → (𝐹‘𝑎) = ∅) | |
8 | 6, 7 | sylnbi 330 | . . 3 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → (𝐹‘𝑎) = ∅) |
9 | 2, 8 | eqtr4d 2770 | . 2 ⊢ (¬ 𝑎 ∈ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦} → ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎)) |
10 | 1, 9 | pm2.61i 182 | 1 ⊢ ((𝐹 ↾ {𝑥 ∣ ∃!𝑦 𝑥𝐹𝑦})‘𝑎) = (𝐹‘𝑎) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1534 ∈ wcel 2099 ∃!weu 2557 {cab 2704 ∅c0 4318 class class class wbr 5142 ↾ cres 5674 ‘cfv 6542 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-dm 5682 df-res 5684 df-iota 6494 df-fv 6550 |
This theorem is referenced by: setrec2 48039 |
Copyright terms: Public domain | W3C validator |