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Theorem funprgOLD 5150
 Description: A set of two pairs is a function if their first members are different. (Contributed by FL, 26-Jun-2011.)
Assertion
Ref Expression
funprgOLD ((AB (A V B W) (C T D U)) → Fun {A, C, B, D})

Proof of Theorem funprgOLD
StepHypRef Expression
1 simp2l 981 . . . 4 ((AB (A V B W) (C T D U)) → A V)
2 simp3l 983 . . . 4 ((AB (A V B W) (C T D U)) → C T)
3 funsngOLD 5148 . . . 4 ((A V C T) → Fun {A, C})
41, 2, 3syl2anc 642 . . 3 ((AB (A V B W) (C T D U)) → Fun {A, C})
5 simp2r 982 . . . 4 ((AB (A V B W) (C T D U)) → B W)
6 simp3r 984 . . . 4 ((AB (A V B W) (C T D U)) → D U)
7 funsngOLD 5148 . . . 4 ((B W D U) → Fun {B, D})
85, 6, 7syl2anc 642 . . 3 ((AB (A V B W) (C T D U)) → Fun {B, D})
9 dmsnopg 5066 . . . . . 6 (C T → dom {A, C} = {A})
102, 9syl 15 . . . . 5 ((AB (A V B W) (C T D U)) → dom {A, C} = {A})
11 dmsnopg 5066 . . . . . 6 (D U → dom {B, D} = {B})
126, 11syl 15 . . . . 5 ((AB (A V B W) (C T D U)) → dom {B, D} = {B})
1310, 12ineq12d 3458 . . . 4 ((AB (A V B W) (C T D U)) → (dom {A, C} ∩ dom {B, D}) = ({A} ∩ {B}))
14 disjsn2 3787 . . . . 5 (AB → ({A} ∩ {B}) = )
15143ad2ant1 976 . . . 4 ((AB (A V B W) (C T D U)) → ({A} ∩ {B}) = )
1613, 15eqtrd 2385 . . 3 ((AB (A V B W) (C T D U)) → (dom {A, C} ∩ dom {B, D}) = )
17 funun 5146 . . 3 (((Fun {A, C} Fun {B, D}) (dom {A, C} ∩ dom {B, D}) = ) → Fun ({A, C} ∪ {B, D}))
184, 8, 16, 17syl21anc 1181 . 2 ((AB (A V B W) (C T D U)) → Fun ({A, C} ∪ {B, D}))
19 df-pr 3742 . . 3 {A, C, B, D} = ({A, C} ∪ {B, D})
2019funeqi 5128 . 2 (Fun {A, C, B, D} ↔ Fun ({A, C} ∪ {B, D}))
2118, 20sylibr 203 1 ((AB (A V B W) (C T D U)) → Fun {A, C, B, D})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∧ 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-1 5  ax-2 6  ax-3 7  ax-mp 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:  fnprg  5153
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