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 42230 | . 2 ⊢ setrecs(𝐹) = ∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} | |
2 | nfv 1830 | . . . . . . . 8 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑦 | |
3 | nfv 1830 | . . . . . . . . 9 ⊢ Ⅎ𝑥 𝑤 ⊆ 𝑧 | |
4 | nfsetrecs.1 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹 | |
5 | nfcv 2751 | . . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑤 | |
6 | 4, 5 | nffv 6110 | . . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑤) |
7 | nfcv 2751 | . . . . . . . . . 10 ⊢ Ⅎ𝑥𝑧 | |
8 | 6, 7 | nfss 3561 | . . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑤) ⊆ 𝑧 |
9 | 3, 8 | nfim 1813 | . . . . . . . 8 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧) |
10 | 2, 9 | nfim 1813 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
11 | 10 | nfal 2139 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) |
12 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝑧 | |
13 | 11, 12 | nfim 1813 | . . . . 5 ⊢ Ⅎ𝑥(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
14 | 13 | nfal 2139 | . . . 4 ⊢ Ⅎ𝑥∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧) |
15 | 14 | nfab 2755 | . . 3 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
16 | 15 | nfuni 4378 | . 2 ⊢ Ⅎ𝑥∪ {𝑦 ∣ ∀𝑧(∀𝑤(𝑤 ⊆ 𝑦 → (𝑤 ⊆ 𝑧 → (𝐹‘𝑤) ⊆ 𝑧)) → 𝑦 ⊆ 𝑧)} |
17 | 1, 16 | nfcxfr 2749 | 1 ⊢ Ⅎ𝑥setrecs(𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 {cab 2596 Ⅎwnfc 2738 ⊆ wss 3540 ∪ cuni 4372 ‘cfv 5804 setrecscsetrecs 42229 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-setrecs 42230 |
This theorem is referenced by: (None) |
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