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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 8896 | . 2 ⊢ X𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)} | |
2 | nfcv 2901 | . . . . 5 ⊢ Ⅎ𝑥𝑦 | |
3 | nfab1 2903 | . . . . 5 ⊢ Ⅎ𝑥{𝑥 ∣ 𝑥 ∈ 𝐴} | |
4 | 2, 3 | nffn 6649 | . . . 4 ⊢ Ⅎ𝑥 𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} |
5 | nfra1 3279 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵 | |
6 | 4, 5 | nfan 1900 | . . 3 ⊢ Ⅎ𝑥(𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵) |
7 | 6 | nfab 2907 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ (𝑦 Fn {𝑥 ∣ 𝑥 ∈ 𝐴} ∧ ∀𝑥 ∈ 𝐴 (𝑦‘𝑥) ∈ 𝐵)} |
8 | 1, 7 | nfcxfr 2899 | 1 ⊢ Ⅎ𝑥X𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∈ wcel 2104 {cab 2707 Ⅎwnfc 2881 ∀wral 3059 Fn wfn 6539 ‘cfv 6544 Xcixp 8895 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ral 3060 df-rab 3431 df-v 3474 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 8896 |
This theorem is referenced by: ixpiunwdom 9589 ptbasfi 23307 hoidmvlelem3 45613 hspdifhsp 45632 hoiqssbllem2 45639 hspmbllem2 45643 opnvonmbllem2 45649 iinhoiicc 45690 iunhoiioo 45692 vonioo 45698 vonicc 45701 |
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