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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 2374. See nfiing 5052 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 5022 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfiun.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfiun.2 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2895 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | 2, 4 | nfralw 3312 | . . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 |
6 | 5 | nfab 2910 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2103 {cab 2711 Ⅎwnfc 2888 ∀wral 3063 ∩ ciin 5020 |
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 2105 ax-9 2113 ax-10 2136 ax-11 2153 ax-12 2173 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2890 df-ral 3064 df-iin 5022 |
This theorem is referenced by: iinab 5094 fnlimcnv 45522 fnlimfvre 45529 fnlimabslt 45534 iinhoiicc 46529 preimageiingt 46575 preimaleiinlt 46576 smflimlem6 46631 smflim 46632 smflim2 46661 smfsup 46669 smfsupmpt 46670 smfsupxr 46671 smfinflem 46672 smfinf 46673 smfinfmpt 46674 smflimsup 46683 smfliminf 46686 fsupdm 46697 finfdm 46701 |
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