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Mirrors > Home > NFE Home > Th. List > fpr | GIF version |
Description: A function with a domain of two elements. (Contributed by Jeff Madsen, 20-Jun-2010.) (The proof was shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
fpr.1 | ⊢ A ∈ V |
fpr.2 | ⊢ B ∈ V |
fpr.3 | ⊢ C ∈ V |
fpr.4 | ⊢ D ∈ V |
Ref | Expression |
---|---|
fpr | ⊢ (A ≠ B → {〈A, C〉, 〈B, D〉}:{A, B}–→{C, D}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fpr.3 | . . . . . 6 ⊢ C ∈ V | |
2 | fpr.4 | . . . . . 6 ⊢ D ∈ V | |
3 | 1, 2 | funpr 5151 | . . . . 5 ⊢ (A ≠ B → Fun {〈A, C〉, 〈B, D〉}) |
4 | 1, 2 | dmprop 5070 | . . . . 5 ⊢ dom {〈A, C〉, 〈B, D〉} = {A, B} |
5 | 3, 4 | jctir 524 | . . . 4 ⊢ (A ≠ B → (Fun {〈A, C〉, 〈B, D〉} ∧ dom {〈A, C〉, 〈B, D〉} = {A, B})) |
6 | df-fn 4790 | . . . 4 ⊢ ({〈A, C〉, 〈B, D〉} Fn {A, B} ↔ (Fun {〈A, C〉, 〈B, D〉} ∧ dom {〈A, C〉, 〈B, D〉} = {A, B})) | |
7 | 5, 6 | sylibr 203 | . . 3 ⊢ (A ≠ B → {〈A, C〉, 〈B, D〉} Fn {A, B}) |
8 | fpr.1 | . . . . . . 7 ⊢ A ∈ V | |
9 | 8 | rnsnop 5075 | . . . . . 6 ⊢ ran {〈A, C〉} = {C} |
10 | fpr.2 | . . . . . . 7 ⊢ B ∈ V | |
11 | 10 | rnsnop 5075 | . . . . . 6 ⊢ ran {〈B, D〉} = {D} |
12 | 9, 11 | uneq12i 3416 | . . . . 5 ⊢ (ran {〈A, C〉} ∪ ran {〈B, D〉}) = ({C} ∪ {D}) |
13 | df-pr 3742 | . . . . . . 7 ⊢ {〈A, C〉, 〈B, D〉} = ({〈A, C〉} ∪ {〈B, D〉}) | |
14 | 13 | rneqi 4957 | . . . . . 6 ⊢ ran {〈A, C〉, 〈B, D〉} = ran ({〈A, C〉} ∪ {〈B, D〉}) |
15 | rnun 5036 | . . . . . 6 ⊢ ran ({〈A, C〉} ∪ {〈B, D〉}) = (ran {〈A, C〉} ∪ ran {〈B, D〉}) | |
16 | 14, 15 | eqtri 2373 | . . . . 5 ⊢ ran {〈A, C〉, 〈B, D〉} = (ran {〈A, C〉} ∪ ran {〈B, D〉}) |
17 | df-pr 3742 | . . . . 5 ⊢ {C, D} = ({C} ∪ {D}) | |
18 | 12, 16, 17 | 3eqtr4i 2383 | . . . 4 ⊢ ran {〈A, C〉, 〈B, D〉} = {C, D} |
19 | 18 | eqimssi 3325 | . . 3 ⊢ ran {〈A, C〉, 〈B, D〉} ⊆ {C, D} |
20 | 7, 19 | jctir 524 | . 2 ⊢ (A ≠ B → ({〈A, C〉, 〈B, D〉} Fn {A, B} ∧ ran {〈A, C〉, 〈B, D〉} ⊆ {C, D})) |
21 | df-f 4791 | . 2 ⊢ ({〈A, C〉, 〈B, D〉}:{A, B}–→{C, D} ↔ ({〈A, C〉, 〈B, D〉} Fn {A, B} ∧ ran {〈A, C〉, 〈B, D〉} ⊆ {C, D})) | |
22 | 20, 21 | sylibr 203 | 1 ⊢ (A ≠ B → {〈A, C〉, 〈B, D〉}:{A, B}–→{C, D}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 Vcvv 2859 ∪ cun 3207 ⊆ wss 3257 {csn 3737 {cpr 3738 〈cop 4561 dom cdm 4772 ran crn 4773 Fun wfun 4775 Fn wfn 4776 –→wf 4777 |
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-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-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-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 |
This theorem is referenced by: (None) |
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