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| Mirrors > Home > MPE Home > Th. List > Mathboxes > nfsetrecs | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for setrecs. (Contributed by Emmett Weisz, 21-Oct-2021.) |
| Ref | Expression |
|---|---|
| nfsetrecs.1 | ⊢ Ⅎ𝑥𝐹 |
| Ref | Expression |
|---|---|
| nfsetrecs | ⊢ Ⅎ𝑥setrecs(𝐹) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-setrecs 49203 | . 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
| 2 | nfv 1914 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦 | |
| 3 | nfv 1914 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧 | |
| 4 | nfsetrecs.1 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
| 5 | nfcv 2905 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤 | |
| 6 | 4, 5 | nffv 6916 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤) |
| 7 | nfcv 2905 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
| 8 | 6, 7 | nfss 3976 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧 |
| 9 | 3, 8 | nfim 1896 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) |
| 10 | 2, 9 | nfim 1896 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
| 11 | 10 | nfal 2323 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
| 12 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧 | |
| 13 | 11, 12 | nfim 1896 | . . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
| 14 | 13 | nfal 2323 | . . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
| 15 | 14 | nfab 2911 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
| 16 | 15 | nfuni 4914 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
| 17 | 1, 16 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥setrecs(𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 {cab 2714 Ⅎwnfc 2890 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 setrecscsetrecs 49202 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-setrecs 49203 |
| This theorem is referenced by: (None) |
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