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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 8598 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by Mario Carneiro, 15-Oct-2016.) (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 8579 | . 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 1550 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
9 | 2, 8 | nffn 6478 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
10 | df-ral 3066 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | |
11 | nftru 1812 | . . . . . . 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 6729 | . . . . . . . . 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 1902 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
20 | 11, 19 | nfald 2327 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
21 | 20 | mptru 1550 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) |
22 | 10, 21 | nfxfr 1860 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 |
23 | 9, 22 | nfan 1907 | . . 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 399 ∀wal 1541 ⊤wtru 1544 Ⅎwnf 1791 ∈ wcel 2110 {cab 2714 Ⅎwnfc 2884 ∀wral 3061 Fn wfn 6375 ‘cfv 6380 Xcixp 8578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-iota 6338 df-fun 6382 df-fn 6383 df-fv 6388 df-ixp 8579 |
This theorem is referenced by: vonioo 43895 |
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