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Mirrors > Home > ILE Home > Th. List > nfixpxy | GIF version |
Description: Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.) |
Ref | Expression |
---|---|
nfixp.1 | ⊢ Ⅎ𝑦𝐴 |
nfixp.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfixpxy | ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ixp 6656 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2306 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
3 | nftru 1453 | . . . . . . 7 ⊢ Ⅎ𝑥⊤ | |
4 | nfcvd 2307 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝑥) | |
5 | nfixp.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
6 | 5 | a1i 9 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝐴) |
7 | 4, 6 | nfeld 2322 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴) |
8 | 3, 7 | nfabd 2326 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) |
9 | 8 | mptru 1351 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
10 | 2, 9 | nffn 5278 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
11 | df-ral 2447 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | |
12 | 2 | a1i 9 | . . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧) |
13 | 12, 4 | nffvd 5492 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥)) |
14 | nfixp.2 | . . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵 | |
15 | 14 | a1i 9 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵) |
16 | 13, 15 | nfeld 2322 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) |
17 | 7, 16 | nfimd 1572 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
18 | 3, 17 | nfald 1747 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
19 | 18 | mptru 1351 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) |
20 | 11, 19 | nfxfr 1461 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 |
21 | 10, 20 | nfan 1552 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) |
22 | 21 | nfab 2311 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} |
23 | 1, 22 | nfcxfr 2303 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1340 ⊤wtru 1343 Ⅎwnf 1447 ∈ wcel 2135 {cab 2150 Ⅎwnfc 2293 ∀wral 2442 Fn wfn 5177 ‘cfv 5182 Xcixp 6655 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-iota 5147 df-fun 5184 df-fn 5185 df-fv 5190 df-ixp 6656 |
This theorem is referenced by: (None) |
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