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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 8481 with a disjoint variable condition, which does not require ax-13 2390. (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 8462 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
3 | nfcv 2977 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝑥 | |
4 | nfixpw.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
5 | 3, 4 | nfel 2992 | . . . . . . . 8 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴 |
6 | 5 | nfab 2984 | . . . . . . 7 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
7 | 6 | a1i 11 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) |
8 | 7 | mptru 1544 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
9 | 2, 8 | nffn 6452 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
10 | df-ral 3143 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | |
11 | nftru 1805 | . . . . . . 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 6682 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥)) |
16 | nfixpw.2 | . . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵 | |
17 | 16 | a1i 11 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵) |
18 | 15, 17 | nfeld 2989 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) |
19 | 12, 18 | nfimd 1895 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
20 | 11, 19 | nfald 2347 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
21 | 20 | mptru 1544 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) |
22 | 10, 21 | nfxfr 1853 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 |
23 | 9, 22 | nfan 1900 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) |
24 | 23 | nfab 2984 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} |
25 | 1, 24 | nfcxfr 2975 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∀wal 1535 ⊤wtru 1538 Ⅎwnf 1784 ∈ wcel 2114 {cab 2799 Ⅎwnfc 2961 ∀wral 3138 Fn wfn 6350 ‘cfv 6355 Xcixp 8461 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-ixp 8462 |
This theorem is referenced by: vonioo 43013 |
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