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Mirrors > Home > MPE Home > Th. List > strlemor2OLD | Structured version Visualization version GIF version |
Description: Add two elements to the end of a structure. Obsolete as of 26-Nov-2021. See comment of strlemor0OLD 15968. (Contributed by Stefan O'Rear, 3-Jan-2015.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
strlemor.f | ⊢ (Fun ◡◡𝐹 ∧ dom 𝐹 ⊆ (1...𝐼)) |
strlemor.i | ⊢ 𝐼 ∈ ℕ0 |
strlemor.o | ⊢ 𝐼 < 𝐽 |
strlemor.j | ⊢ 𝐽 ∈ ℕ |
strlemor.a | ⊢ 𝐴 = 𝐽 |
strlemor2.o | ⊢ 𝐽 < 𝐾 |
strlemor2.k | ⊢ 𝐾 ∈ ℕ |
strlemor2.b | ⊢ 𝐵 = 𝐾 |
strlemor2.g | ⊢ 𝐺 = (𝐹 ∪ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉}) |
Ref | Expression |
---|---|
strlemor2OLD | ⊢ (Fun ◡◡𝐺 ∧ dom 𝐺 ⊆ (1...𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | strlemor.f | . . 3 ⊢ (Fun ◡◡𝐹 ∧ dom 𝐹 ⊆ (1...𝐼)) | |
2 | strlemor.i | . . 3 ⊢ 𝐼 ∈ ℕ0 | |
3 | strlemor.o | . . 3 ⊢ 𝐼 < 𝐽 | |
4 | strlemor.j | . . 3 ⊢ 𝐽 ∈ ℕ | |
5 | strlemor.a | . . 3 ⊢ 𝐴 = 𝐽 | |
6 | eqid 2622 | . . 3 ⊢ (𝐹 ∪ {〈𝐴, 𝑋〉}) = (𝐹 ∪ {〈𝐴, 𝑋〉}) | |
7 | 1, 2, 3, 4, 5, 6 | strlemor1OLD 15969 | . 2 ⊢ (Fun ◡◡(𝐹 ∪ {〈𝐴, 𝑋〉}) ∧ dom (𝐹 ∪ {〈𝐴, 𝑋〉}) ⊆ (1...𝐽)) |
8 | 4 | nnnn0i 11300 | . 2 ⊢ 𝐽 ∈ ℕ0 |
9 | strlemor2.o | . 2 ⊢ 𝐽 < 𝐾 | |
10 | strlemor2.k | . 2 ⊢ 𝐾 ∈ ℕ | |
11 | strlemor2.b | . 2 ⊢ 𝐵 = 𝐾 | |
12 | df-pr 4180 | . . . 4 ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} = ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉}) | |
13 | 12 | uneq2i 3764 | . . 3 ⊢ (𝐹 ∪ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉}) = (𝐹 ∪ ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉})) |
14 | strlemor2.g | . . 3 ⊢ 𝐺 = (𝐹 ∪ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉}) | |
15 | unass 3770 | . . 3 ⊢ ((𝐹 ∪ {〈𝐴, 𝑋〉}) ∪ {〈𝐵, 𝑌〉}) = (𝐹 ∪ ({〈𝐴, 𝑋〉} ∪ {〈𝐵, 𝑌〉})) | |
16 | 13, 14, 15 | 3eqtr4i 2654 | . 2 ⊢ 𝐺 = ((𝐹 ∪ {〈𝐴, 𝑋〉}) ∪ {〈𝐵, 𝑌〉}) |
17 | 7, 8, 9, 10, 11, 16 | strlemor1OLD 15969 | 1 ⊢ (Fun ◡◡𝐺 ∧ dom 𝐺 ⊆ (1...𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 {csn 4177 {cpr 4179 〈cop 4183 class class class wbr 4653 ◡ccnv 5113 dom cdm 5114 Fun wfun 5882 (class class class)co 6650 1c1 9937 < clt 10074 ℕcn 11020 ℕ0cn0 11292 ...cfz 12326 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
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-nel 2898 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 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-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 |
This theorem is referenced by: strlemor3OLD 15971 |
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