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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 48003 | . 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
2 | nfv 1909 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦 | |
3 | nfv 1909 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧 | |
4 | nfsetrecs.1 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2897 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤 | |
6 | 4, 5 | nffv 6895 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤) |
7 | nfcv 2897 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 7 | nfss 3969 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧 |
9 | 3, 8 | nfim 1891 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) |
10 | 2, 9 | nfim 1891 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
11 | 10 | nfal 2310 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
12 | nfv 1909 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧 | |
13 | 11, 12 | nfim 1891 | . . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
14 | 13 | nfal 2310 | . . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
15 | 14 | nfab 2903 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
16 | 15 | nfuni 4909 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
17 | 1, 16 | nfcxfr 2895 | 1 ⊢ Ⅎ𝑥setrecs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1531 {cab 2703 Ⅎwnfc 2877 ⊆ wss 3943 ∪ cuni 4902 ‘cfv 6537 setrecscsetrecs 48002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-iota 6489 df-fv 6545 df-setrecs 48003 |
This theorem is referenced by: (None) |
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