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Mirrors > Home > MPE Home > Th. List > Mathboxes > nofnbday | Structured version Visualization version GIF version |
Description: A surreal is a function over its birthday. (Contributed by Scott Fenton, 16-Jun-2011.) |
Ref | Expression |
---|---|
nofnbday | ⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nofun 31046 | . 2 ⊢ (𝐴 ∈ No → Fun 𝐴) | |
2 | bdayval 31045 | . . 3 ⊢ (𝐴 ∈ No → ( bday ‘𝐴) = dom 𝐴) | |
3 | 2 | eqcomd 2616 | . 2 ⊢ (𝐴 ∈ No → dom 𝐴 = ( bday ‘𝐴)) |
4 | df-fn 5807 | . 2 ⊢ (𝐴 Fn ( bday ‘𝐴) ↔ (Fun 𝐴 ∧ dom 𝐴 = ( bday ‘𝐴))) | |
5 | 1, 3, 4 | sylanbrc 695 | 1 ⊢ (𝐴 ∈ No → 𝐴 Fn ( bday ‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 ‘cfv 5804 No csur 31037 bday cbday 31039 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-no 31040 df-bday 31042 |
This theorem is referenced by: nodenselem4 31083 nodenselem6 31085 nodenselem8 31087 |
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