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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 8892 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfab1 2906 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} | |
4 | 2, 3 | nffn 6649 | . . . 4 ⊢ Ⅎ𝑥 𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
5 | nfra1 3282 | . . . 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 2910 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 ∀wral 3062 Fn wfn 6539 ‘cfv 6544 Xcixp 8891 |
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 2704 |
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 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-fun 6546 df-fn 6547 df-ixp 8892 |
This theorem is referenced by: ixpiunwdom 9585 ptbasfi 23085 hoidmvlelem3 45361 hspdifhsp 45380 hoiqssbllem2 45387 hspmbllem2 45391 opnvonmbllem2 45397 iinhoiicc 45438 iunhoiioo 45440 vonioo 45446 vonicc 45449 |
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