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Mirrors > Home > MPE Home > Th. List > Mathboxes > elno3 | Structured version Visualization version GIF version |
Description: Another condition for membership in No . (Contributed by Scott Fenton, 14-Apr-2012.) |
Ref | Expression |
---|---|
elno3 | ⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anan32 1050 | . 2 ⊢ ((Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) ∧ dom 𝐴 ∈ On)) | |
2 | elno2 31807 | . 2 ⊢ (𝐴 ∈ No ↔ (Fun 𝐴 ∧ dom 𝐴 ∈ On ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) | |
3 | df-f 5892 | . . . 4 ⊢ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) | |
4 | funfn 5918 | . . . . 5 ⊢ (Fun 𝐴 ↔ 𝐴 Fn dom 𝐴) | |
5 | 4 | anbi1i 731 | . . . 4 ⊢ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) ↔ (𝐴 Fn dom 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) |
6 | 3, 5 | bitr4i 267 | . . 3 ⊢ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ↔ (Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜})) |
7 | 6 | anbi1i 731 | . 2 ⊢ ((𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On) ↔ ((Fun 𝐴 ∧ ran 𝐴 ⊆ {1𝑜, 2𝑜}) ∧ dom 𝐴 ∈ On)) |
8 | 1, 2, 7 | 3bitr4i 292 | 1 ⊢ (𝐴 ∈ No ↔ (𝐴:dom 𝐴⟶{1𝑜, 2𝑜} ∧ dom 𝐴 ∈ On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∧ wa 384 ∧ w3a 1037 ∈ wcel 1990 ⊆ wss 3574 {cpr 4179 dom cdm 5114 ran crn 5115 Oncon0 5723 Fun wfun 5882 Fn wfn 5883 ⟶wf 5884 1𝑜c1o 7553 2𝑜c2o 7554 No csur 31793 |
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 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-no 31796 |
This theorem is referenced by: noxp1o 31816 noseponlem 31817 |
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