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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.) |
Ref | Expression |
---|---|
nfiun.1 | ⊢ Ⅎ𝑦𝐴 |
nfiun.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfiin | ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 4523 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfiun.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfiun.2 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2758 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | 2, 4 | nfral 2945 | . . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 |
6 | 5 | nfab 2769 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2762 | 1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1990 {cab 2608 Ⅎwnfc 2751 ∀wral 2912 ∩ ciin 4521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-iin 4523 |
This theorem is referenced by: iinab 4581 fnlimcnv 39899 fnlimfvre 39906 fnlimabslt 39911 iinhoiicc 40888 preimageiingt 40930 preimaleiinlt 40931 smflimlem6 40984 smflim 40985 smflim2 41012 smfsup 41020 smfsupmpt 41021 smfsupxr 41022 smfinflem 41023 smfinf 41024 smfinfmpt 41025 smflimsup 41034 smfliminf 41037 |
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