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Mirrors > Home > MPE Home > Th. List > nfixpw | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for indexed Cartesian product. Version of nfixp 8858 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 15-Oct-2016.) Avoid ax-13 2371. (Revised by Gino Giotto, 26-Jan-2024.) |
Ref | Expression |
---|---|
nfixpw.1 | ⊢ Ⅎ𝑦𝐴 |
nfixpw.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfixpw | ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ixp 8839 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
3 | nfcv 2904 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝑥 | |
4 | nfixpw.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
5 | 3, 4 | nfel 2918 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
6 | 5 | nfab 2910 | . . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) |
8 | 7 | mptru 1549 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
9 | 2, 8 | nffn 6602 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
10 | df-ral 3062 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | |
11 | nftru 1807 | . . . . . . 7 ⊢ Ⅎ𝑥⊤ | |
12 | 5 | a1i 11 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴) |
13 | 2 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧) |
14 | 3 | a1i 11 | . . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑥) |
15 | 13, 14 | nffvd 6855 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥)) |
16 | nfixpw.2 | . . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵 | |
17 | 16 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵) |
18 | 15, 17 | nfeld 2915 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) |
19 | 12, 18 | nfimd 1898 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
20 | 11, 19 | nfald 2322 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
21 | 20 | mptru 1549 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) |
22 | 10, 21 | nfxfr 1856 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 |
23 | 9, 22 | nfan 1903 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) |
24 | 23 | nfab 2910 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} |
25 | 1, 24 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∀wal 1540 ⊤wtru 1543 Ⅎwnf 1786 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 ∀wral 3061 Fn wfn 6492 ‘cfv 6497 Xcixp 8838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 df-ixp 8839 |
This theorem is referenced by: vonioo 45009 |
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