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Mirrors > Home > MPE Home > Th. List > nfiin | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) Add disjoint variable condition to avoid ax-13 2380. See nfiing 5049 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) |
Ref | Expression |
---|---|
nfiun.1 | ⊢ Ⅎ𝑦𝐴 |
nfiun.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfiin | ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 5018 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfiun.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfiun.2 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2900 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | 2, 4 | nfralw 3317 | . . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 |
6 | 5 | nfab 2914 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2906 | 1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2108 {cab 2717 Ⅎwnfc 2893 ∀wral 3067 ∩ ciin 5016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-iin 5018 |
This theorem is referenced by: iinab 5091 fnlimcnv 45588 fnlimfvre 45595 fnlimabslt 45600 iinhoiicc 46595 preimageiingt 46641 preimaleiinlt 46642 smflimlem6 46697 smflim 46698 smflim2 46727 smfsup 46735 smfsupmpt 46736 smfsupxr 46737 smfinflem 46738 smfinf 46739 smfinfmpt 46740 smflimsup 46749 smfliminf 46752 fsupdm 46763 finfdm 46767 |
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