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| Description: Bound-variable hypothesis builder for indexed Cartesian product. Usage of this theorem is discouraged because it depends on ax-13 2376. Use the weaker nfixpw 8957 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 8939 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
| 2 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
| 3 | nftru 1803 | . . . . . . 7 ⊢ Ⅎ𝑥⊤ | |
| 4 | nfcvf 2931 | . . . . . . . . 9 ⊢ (¬ ∀𝑦 𝑦 = 𝑥 → Ⅎ𝑦𝑥) | |
| 5 | 4 | adantl 481 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝑥) | 
| 6 | nfixp.1 | . . . . . . . . 9 ⊢ Ⅎ𝑦𝐴 | |
| 7 | 6 | a1i 11 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦𝐴) | 
| 8 | 5, 7 | nfeld 2916 | . . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦 𝑥 ∈ 𝐴) | 
| 9 | 3, 8 | nfabd2 2928 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) | 
| 10 | 9 | mptru 1546 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} | 
| 11 | 2, 10 | nffn 6666 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} | 
| 12 | df-ral 3061 | . . . . 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 6917 | . . . . . . . . 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 2916 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) | 
| 18 | 8, 17 | nfimd 1893 | . . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | 
| 19 | 3, 18 | nfald2 2449 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | 
| 20 | 19 | mptru 1546 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) | 
| 21 | 12, 20 | nfxfr 1852 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 | 
| 22 | 11, 21 | nfan 1898 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) | 
| 23 | 22 | nfab 2910 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | 
| 24 | 1, 23 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1537 ⊤wtru 1540 Ⅎwnf 1782 ∈ wcel 2107 {cab 2713 Ⅎwnfc 2889 ∀wral 3060 Fn wfn 6555 ‘cfv 6560 Xcixp 8938 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-13 2376 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-iota 6513 df-fun 6562 df-fn 6563 df-fv 6568 df-ixp 8939 | 
| This theorem is referenced by: (None) | 
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