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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 2377. Use the weaker nfixpw 8956 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 8938 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | |
| 2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑦𝑧 | |
| 3 | nftru 1804 | . . . . . . 7 ⊢ Ⅎ𝑥⊤ | |
| 4 | nfcvf 2932 | . . . . . . . . 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 2917 | . . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦 𝑥 ∈ 𝐴) | 
| 9 | 3, 8 | nfabd2 2929 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴}) | 
| 10 | 9 | mptru 1547 | . . . . 5 ⊢ Ⅎ𝑦{𝑥 ∣ 𝑥 ∈ 𝐴} | 
| 11 | 2, 10 | nffn 6667 | . . . 4 ⊢ Ⅎ𝑦 𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} | 
| 12 | df-ral 3062 | . . . . 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 6918 | . . . . . . . . 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 2917 | . . . . . . . 8 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑧‘𝑥) ∈ 𝐵) | 
| 18 | 8, 17 | nfimd 1894 | . . . . . . 7 ⊢ ((⊤ ∧ ¬ ∀𝑦 𝑦 = 𝑥) → Ⅎ𝑦(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | 
| 19 | 3, 18 | nfald2 2450 | . . . . . 6 ⊢ (⊤ → Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵)) | 
| 20 | 19 | mptru 1547 | . . . . 5 ⊢ Ⅎ𝑦∀𝑥(𝑥 ∈ 𝐴 → (𝑧‘𝑥) ∈ 𝐵) | 
| 21 | 12, 20 | nfxfr 1853 | . . . 4 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵 | 
| 22 | 11, 21 | nfan 1899 | . . 3 ⊢ Ⅎ𝑦(𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵) | 
| 23 | 22 | nfab 2911 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ (𝑧 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑧‘𝑥) ∈ 𝐵)} | 
| 24 | 1, 23 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑦X𝑥 ∈ 𝐴 𝐵 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1538 ⊤wtru 1541 Ⅎwnf 1783 ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 ∀wral 3061 Fn wfn 6556 ‘cfv 6561 Xcixp 8937 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ixp 8938 | 
| This theorem is referenced by: (None) | 
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