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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 47101 | . 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
2 | nfv 1917 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦 | |
3 | nfv 1917 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧 | |
4 | nfsetrecs.1 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2907 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤 | |
6 | 4, 5 | nffv 6852 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤) |
7 | nfcv 2907 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 7 | nfss 3936 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧 |
9 | 3, 8 | nfim 1899 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) |
10 | 2, 9 | nfim 1899 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
11 | 10 | nfal 2316 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
12 | nfv 1917 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧 | |
13 | 11, 12 | nfim 1899 | . . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
14 | 13 | nfal 2316 | . . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
15 | 14 | nfab 2913 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
16 | 15 | nfuni 4872 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
17 | 1, 16 | nfcxfr 2905 | 1 ⊢ Ⅎ𝑥setrecs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1539 {cab 2713 Ⅎwnfc 2887 ⊆ wss 3910 ∪ cuni 4865 ‘cfv 6496 setrecscsetrecs 47100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-iota 6448 df-fv 6504 df-setrecs 47101 |
This theorem is referenced by: (None) |
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