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Mirrors > Home > NFE Home > Th. List > funprg | GIF version |
Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.) (Revised by Scott Fenton, 16-Apr-2021.) |
Ref | Expression |
---|---|
funprg | ⊢ ((A ≠ B ∧ C ∈ V ∧ D ∈ W) → Fun {〈A, C〉, 〈B, D〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmsnopg 5066 | . . . . . 6 ⊢ (C ∈ V → dom {〈A, C〉} = {A}) | |
2 | 1 | 3ad2ant2 977 | . . . . 5 ⊢ ((A ≠ B ∧ C ∈ V ∧ D ∈ W) → dom {〈A, C〉} = {A}) |
3 | dmsnopg 5066 | . . . . . 6 ⊢ (D ∈ W → dom {〈B, D〉} = {B}) | |
4 | 3 | 3ad2ant3 978 | . . . . 5 ⊢ ((A ≠ B ∧ C ∈ V ∧ D ∈ W) → dom {〈B, D〉} = {B}) |
5 | 2, 4 | ineq12d 3458 | . . . 4 ⊢ ((A ≠ B ∧ C ∈ V ∧ D ∈ W) → (dom {〈A, C〉} ∩ dom {〈B, D〉}) = ({A} ∩ {B})) |
6 | disjsn2 3787 | . . . . 5 ⊢ (A ≠ B → ({A} ∩ {B}) = ∅) | |
7 | 6 | 3ad2ant1 976 | . . . 4 ⊢ ((A ≠ B ∧ C ∈ V ∧ D ∈ W) → ({A} ∩ {B}) = ∅) |
8 | 5, 7 | eqtrd 2385 | . . 3 ⊢ ((A ≠ B ∧ C ∈ V ∧ D ∈ W) → (dom {〈A, C〉} ∩ dom {〈B, D〉}) = ∅) |
9 | funsn 5147 | . . . 4 ⊢ Fun {〈A, C〉} | |
10 | funsn 5147 | . . . 4 ⊢ Fun {〈B, D〉} | |
11 | funun 5146 | . . . 4 ⊢ (((Fun {〈A, C〉} ∧ Fun {〈B, D〉}) ∧ (dom {〈A, C〉} ∩ dom {〈B, D〉}) = ∅) → Fun ({〈A, C〉} ∪ {〈B, D〉})) | |
12 | 9, 10, 11 | mpanl12 663 | . . 3 ⊢ ((dom {〈A, C〉} ∩ dom {〈B, D〉}) = ∅ → Fun ({〈A, C〉} ∪ {〈B, D〉})) |
13 | 8, 12 | syl 15 | . 2 ⊢ ((A ≠ B ∧ C ∈ V ∧ D ∈ W) → Fun ({〈A, C〉} ∪ {〈B, D〉})) |
14 | df-pr 3742 | . . 3 ⊢ {〈A, C〉, 〈B, D〉} = ({〈A, C〉} ∪ {〈B, D〉}) | |
15 | 14 | funeqi 5128 | . 2 ⊢ (Fun {〈A, C〉, 〈B, D〉} ↔ Fun ({〈A, C〉} ∪ {〈B, D〉})) |
16 | 13, 15 | sylibr 203 | 1 ⊢ ((A ≠ B ∧ C ∈ V ∧ D ∈ W) → Fun {〈A, C〉, 〈B, D〉}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 934 = wceq 1642 ∈ wcel 1710 ≠ wne 2516 ∪ cun 3207 ∩ cin 3208 ∅c0 3550 {csn 3737 {cpr 3738 〈cop 4561 dom cdm 4772 Fun wfun 4775 |
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 |
This theorem is referenced by: funpr 5151 |
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