Mathbox for Emmett Weisz |
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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 44794 | . 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
2 | nfv 1915 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦 | |
3 | nfv 1915 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧 | |
4 | nfsetrecs.1 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2979 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤 | |
6 | 4, 5 | nffv 6682 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤) |
7 | nfcv 2979 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 7 | nfss 3962 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧 |
9 | 3, 8 | nfim 1897 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) |
10 | 2, 9 | nfim 1897 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
11 | 10 | nfal 2342 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
12 | nfv 1915 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧 | |
13 | 11, 12 | nfim 1897 | . . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
14 | 13 | nfal 2342 | . . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
15 | 14 | nfab 2986 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
16 | 15 | nfuni 4847 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
17 | 1, 16 | nfcxfr 2977 | 1 ⊢ Ⅎ𝑥setrecs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1535 {cab 2801 Ⅎwnfc 2963 ⊆ wss 3938 ∪ cuni 4840 ‘cfv 6357 setrecscsetrecs 44793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-setrecs 44794 |
This theorem is referenced by: (None) |
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