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Mirrors > Home > MPE Home > Th. List > Mathboxes > bdayimaon | Structured version Visualization version GIF version |
Description: Lemma for full-eta properties. The successor of the union of the image of the birthday function under a set is an ordinal. (Contributed by Scott Fenton, 20-Aug-2011.) |
Ref | Expression |
---|---|
bdayimaon | ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdayfo 31828 | . . . . . 6 ⊢ bday : No –onto→On | |
2 | fofun 6116 | . . . . . 6 ⊢ ( bday : No –onto→On → Fun bday ) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ Fun bday |
4 | funimaexg 5975 | . . . . 5 ⊢ ((Fun bday ∧ 𝐴 ∈ 𝑉) → ( bday “ 𝐴) ∈ V) | |
5 | 3, 4 | mpan 706 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → ( bday “ 𝐴) ∈ V) |
6 | uniexg 6955 | . . . 4 ⊢ (( bday “ 𝐴) ∈ V → ∪ ( bday “ 𝐴) ∈ V) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ ( bday “ 𝐴) ∈ V) |
8 | imassrn 5477 | . . . . 5 ⊢ ( bday “ 𝐴) ⊆ ran bday | |
9 | forn 6118 | . . . . . 6 ⊢ ( bday : No –onto→On → ran bday = On) | |
10 | 1, 9 | ax-mp 5 | . . . . 5 ⊢ ran bday = On |
11 | 8, 10 | sseqtri 3637 | . . . 4 ⊢ ( bday “ 𝐴) ⊆ On |
12 | ssorduni 6985 | . . . 4 ⊢ (( bday “ 𝐴) ⊆ On → Ord ∪ ( bday “ 𝐴)) | |
13 | 11, 12 | ax-mp 5 | . . 3 ⊢ Ord ∪ ( bday “ 𝐴) |
14 | 7, 13 | jctil 560 | . 2 ⊢ (𝐴 ∈ 𝑉 → (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V)) |
15 | elon2 5734 | . . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ (Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V)) | |
16 | sucelon 7017 | . . 3 ⊢ (∪ ( bday “ 𝐴) ∈ On ↔ suc ∪ ( bday “ 𝐴) ∈ On) | |
17 | 15, 16 | bitr3i 266 | . 2 ⊢ ((Ord ∪ ( bday “ 𝐴) ∧ ∪ ( bday “ 𝐴) ∈ V) ↔ suc ∪ ( bday “ 𝐴) ∈ On) |
18 | 14, 17 | sylib 208 | 1 ⊢ (𝐴 ∈ 𝑉 → suc ∪ ( bday “ 𝐴) ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 ∪ cuni 4436 ran crn 5115 “ cima 5117 Ord word 5722 Oncon0 5723 suc csuc 5725 Fun wfun 5882 –onto→wfo 5886 No csur 31793 bday cbday 31795 |
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-8 1992 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 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-ord 5726 df-on 5727 df-suc 5729 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-1o 7560 df-no 31796 df-bday 31798 |
This theorem is referenced by: noetalem1 31863 |
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