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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 6713 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2329 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
3 | nftru 1476 | . . . . . . 7 ⊢ Ⅎ𝑥⊤ | |
4 | nfcvd 2330 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝑥) | |
5 | nfixp.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
6 | 5 | a1i 9 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦𝐴) |
7 | 4, 6 | nfeld 2345 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦 𝑥 ∈ 𝐴) |
8 | 3, 7 | nfabd 2349 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) |
9 | 8 | mptru 1372 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} |
10 | 2, 9 | nffn 5324 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
11 | df-ral 2470 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | |
12 | 2 | a1i 9 | . . . . . . . . . 10 ⊢ (⊤ → Ⅎ𝑦𝑧) |
13 | 12, 4 | nffvd 5539 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥)) |
14 | nfixp.2 | . . . . . . . . . 10 ⊢ Ⅎ𝑦𝐵 | |
15 | 14 | a1i 9 | . . . . . . . . 9 ⊢ (⊤ → Ⅎ𝑦𝐵) |
16 | 13, 15 | nfeld 2345 | . . . . . . . 8 ⊢ (⊤ → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) |
17 | 7, 16 | nfimd 1595 | . . . . . . 7 ⊢ (⊤ → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
18 | 3, 17 | nfald 1770 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) |
19 | 18 | mptru 1372 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) |
20 | 11, 19 | nfxfr 1484 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 |
21 | 10, 20 | nfan 1575 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) |
22 | 21 | nfab 2334 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} |
23 | 1, 22 | nfcxfr 2326 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∀wal 1361 ⊤wtru 1364 Ⅎwnf 1470 ∈ wcel 2158 {cab 2173 Ⅎwnfc 2316 ∀wral 2465 Fn wfn 5223 ‘cfv 5228 Xcixp 6712 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-iota 5190 df-fun 5230 df-fn 5231 df-fv 5236 df-ixp 6713 |
This theorem is referenced by: (None) |
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