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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 6673 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2312 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
3 | nftru 1459 | . . . . . . 7 ⊢ Ⅎ𝑥⊤ | |
4 | nfcvd 2313 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝑥) | |
5 | nfixp.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
6 | 5 | a1i 9 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝐴) |
7 | 4, 6 | nfeld 2328 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴) |
8 | 3, 7 | nfabd 2332 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) |
9 | 8 | mptru 1357 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
10 | 2, 9 | nffn 5292 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
11 | df-ral 2453 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | |
12 | 2 | a1i 9 | . . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧) |
13 | 12, 4 | nffvd 5506 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥)) |
14 | nfixp.2 | . . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵 | |
15 | 14 | a1i 9 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵) |
16 | 13, 15 | nfeld 2328 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) |
17 | 7, 16 | nfimd 1578 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
18 | 3, 17 | nfald 1753 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
19 | 18 | mptru 1357 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) |
20 | 11, 19 | nfxfr 1467 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 |
21 | 10, 20 | nfan 1558 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) |
22 | 21 | nfab 2317 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} |
23 | 1, 22 | nfcxfr 2309 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∀wal 1346 ⊤wtru 1349 Ⅎwnf 1453 ∈ wcel 2141 {cab 2156 Ⅎwnfc 2299 ∀wral 2448 Fn wfn 5191 ‘cfv 5196 Xcixp 6672 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fn 5199 df-fv 5204 df-ixp 6673 |
This theorem is referenced by: (None) |
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