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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 2365. Use the weaker nfixpw 8935 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 8917 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2891 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
3 | nftru 1798 | . . . . . . 7 ⊢ Ⅎ𝑥⊤ | |
4 | nfcvf 2921 | . . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥) | |
5 | 4 | adantl 480 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝑥) |
6 | nfixp.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
7 | 6 | a1i 11 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝐴) |
8 | 5, 7 | nfeld 2903 | . . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦 𝑥 ∈ 𝐴) |
9 | 3, 8 | nfabd2 2918 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) |
10 | 9 | mptru 1540 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
11 | 2, 10 | nffn 6654 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
12 | df-ral 3051 | . . . . 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 6908 | . . . . . . . . 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 2903 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) |
18 | 8, 17 | nfimd 1889 | . . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
19 | 3, 18 | nfald2 2438 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
20 | 19 | mptru 1540 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) |
21 | 12, 20 | nfxfr 1847 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 |
22 | 11, 21 | nfan 1894 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) |
23 | 22 | nfab 2897 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} |
24 | 1, 23 | nfcxfr 2889 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∀wal 1531 ⊤wtru 1534 Ⅎwnf 1777 ∈ wcel 2098 {cab 2702 Ⅎwnfc 2875 ∀wral 3050 Fn wfn 6544 ‘cfv 6549 Xcixp 8916 |
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 2166 ax-13 2365 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fn 6552 df-fv 6557 df-ixp 8917 |
This theorem is referenced by: (None) |
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