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Mirrors > Home > MPE Home > Th. List > nfixp | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 2375. Use the weaker nfixpw 8955 when possible. (Contributed by Mario Carneiro, 15-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfixp.1 | ⊢ Ⅎ𝑦𝐴 |
nfixp.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfixp | ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ixp 8937 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
3 | nftru 1801 | . . . . . . 7 ⊢ Ⅎ𝑥⊤ | |
4 | nfcvf 2930 | . . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥) | |
5 | 4 | adantl 481 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝑥) |
6 | nfixp.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
7 | 6 | a1i 11 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝐴) |
8 | 5, 7 | nfeld 2915 | . . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦 𝑥 ∈ 𝐴) |
9 | 3, 8 | nfabd2 2927 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) |
10 | 9 | mptru 1544 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
11 | 2, 10 | nffn 6668 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
12 | df-ral 3060 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | |
13 | 2 | a1i 11 | . . . . . . . . . 10 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝑧) |
14 | 13, 5 | nffvd 6919 | . . . . . . . . 9 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧‘𝑥)) |
15 | nfixp.2 | . . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵 | |
16 | 15 | a1i 11 | . . . . . . . . 9 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝐵) |
17 | 14, 16 | nfeld 2915 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) |
18 | 8, 17 | nfimd 1892 | . . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
19 | 3, 18 | nfald2 2448 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
20 | 19 | mptru 1544 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) |
21 | 12, 20 | nfxfr 1850 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 |
22 | 11, 21 | nfan 1897 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) |
23 | 22 | nfab 2909 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} |
24 | 1, 23 | nfcxfr 2901 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1535 ⊤wtru 1538 Ⅎwnf 1780 ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 ∀wral 3059 Fn wfn 6558 ‘cfv 6563 Xcixp 8936 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-ixp 8937 |
This theorem is referenced by: (None) |
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