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Mirrors > Home > MPE Home > Th. List > nfixp1 | Structured version Visualization version GIF version |
Description: The index variable in an indexed Cartesian product is not free. (Contributed by Jeff Madsen, 19-Jun-2011.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nfixp1 | ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ixp 8769 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfab1 2907 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} | |
4 | 2, 3 | nffn 6596 | . . . 4 ⊢ Ⅎ𝑥 𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
5 | nfra1 3265 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵 | |
6 | 4, 5 | nfan 1902 | . . 3 ⊢ Ⅎ𝑥(𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵) |
7 | 6 | nfab 2911 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2106 {cab 2714 Ⅎwnfc 2885 ∀wral 3062 Fn wfn 6486 ‘cfv 6491 Xcixp 8768 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ral 3063 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-br 5104 df-opab 5166 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-fun 6493 df-fn 6494 df-ixp 8769 |
This theorem is referenced by: ixpiunwdom 9459 ptbasfi 22854 hoidmvlelem3 44591 hspdifhsp 44610 hoiqssbllem2 44617 hspmbllem2 44621 opnvonmbllem2 44627 iinhoiicc 44668 iunhoiioo 44670 vonioo 44676 vonicc 44679 |
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