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Mirrors > Home > NFE Home > Th. List > braddcfn | GIF version |
Description: Binary relationship form of the AddC function. (Contributed by SF, 2-Mar-2015.) |
Ref | Expression |
---|---|
braddcfn.1 | ⊢ A ∈ V |
braddcfn.2 | ⊢ B ∈ V |
Ref | Expression |
---|---|
braddcfn | ⊢ (〈A, B〉 AddC C ↔ (A +c B) = C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | addcfn 5825 | . . 3 ⊢ AddC Fn V | |
2 | braddcfn.1 | . . . 4 ⊢ A ∈ V | |
3 | braddcfn.2 | . . . 4 ⊢ B ∈ V | |
4 | 2, 3 | opex 4588 | . . 3 ⊢ 〈A, B〉 ∈ V |
5 | fnbrfvb 5358 | . . 3 ⊢ (( AddC Fn V ∧ 〈A, B〉 ∈ V) → (( AddC ‘〈A, B〉) = C ↔ 〈A, B〉 AddC C)) | |
6 | 1, 4, 5 | mp2an 653 | . 2 ⊢ (( AddC ‘〈A, B〉) = C ↔ 〈A, B〉 AddC C) |
7 | df-ov 5526 | . . . 4 ⊢ (A AddC B) = ( AddC ‘〈A, B〉) | |
8 | addceq1 4383 | . . . . . 6 ⊢ (x = A → (x +c y) = (A +c y)) | |
9 | addceq2 4384 | . . . . . 6 ⊢ (y = B → (A +c y) = (A +c B)) | |
10 | df-addcfn 5746 | . . . . . 6 ⊢ AddC = (x ∈ V, y ∈ V ↦ (x +c y)) | |
11 | 2, 3 | addcex 4394 | . . . . . 6 ⊢ (A +c B) ∈ V |
12 | 8, 9, 10, 11 | ovmpt2 5716 | . . . . 5 ⊢ ((A ∈ V ∧ B ∈ V) → (A AddC B) = (A +c B)) |
13 | 2, 3, 12 | mp2an 653 | . . . 4 ⊢ (A AddC B) = (A +c B) |
14 | 7, 13 | eqtr3i 2375 | . . 3 ⊢ ( AddC ‘〈A, B〉) = (A +c B) |
15 | 14 | eqeq1i 2360 | . 2 ⊢ (( AddC ‘〈A, B〉) = C ↔ (A +c B) = C) |
16 | 6, 15 | bitr3i 242 | 1 ⊢ (〈A, B〉 AddC C ↔ (A +c B) = C) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 ∈ wcel 1710 Vcvv 2859 +c cplc 4375 〈cop 4561 class class class wbr 4639 Fn wfn 4776 ‘cfv 4781 (class class class)co 5525 AddC caddcfn 5745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-csb 3137 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-iun 3971 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-res 4788 df-fun 4789 df-fn 4790 df-f 4791 df-fo 4793 df-fv 4795 df-2nd 4797 df-ov 5526 df-oprab 5528 df-mpt 5652 df-mpt2 5654 df-addcfn 5746 |
This theorem is referenced by: csucex 6259 addccan2nclem1 6263 nncdiv3lem1 6275 nncdiv3lem2 6276 nnc3n3p1 6278 nchoicelem16 6304 |
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