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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 8770 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2906 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfab1 2908 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} | |
4 | 2, 3 | nffn 6597 | . . . 4 ⊢ Ⅎ𝑥 𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
5 | nfra1 3266 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵 | |
6 | 4, 5 | nfan 1903 | . . 3 ⊢ Ⅎ𝑥(𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵) |
7 | 6 | nfab 2912 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2904 | 1 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 {cab 2715 Ⅎwnfc 2886 ∀wral 3063 Fn wfn 6487 ‘cfv 6492 Xcixp 8769 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ral 3064 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-fun 6494 df-fn 6495 df-ixp 8770 |
This theorem is referenced by: ixpiunwdom 9460 ptbasfi 22854 hoidmvlelem3 44560 hspdifhsp 44579 hoiqssbllem2 44586 hspmbllem2 44590 opnvonmbllem2 44596 iinhoiicc 44637 iunhoiioo 44639 vonioo 44645 vonicc 44648 |
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