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Theorem fprg 5289
Description: A function with a domain of two elements. (Contributed by FL, 2-Feb-2014.)
Assertion
Ref Expression
fprg (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → {⟨A, 𝐶⟩, ⟨B, 𝐷⟩}:{A, B}⟶{𝐶, 𝐷})

Proof of Theorem fprg
StepHypRef Expression
1 fnprg 4897 . 2 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} Fn {A, B})
2 rnsnopg 4742 . . . . . . 7 (A 𝐸 → ran {⟨A, 𝐶⟩} = {𝐶})
32adantr 261 . . . . . 6 ((A 𝐸 B 𝐹) → ran {⟨A, 𝐶⟩} = {𝐶})
433ad2ant1 924 . . . . 5 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → ran {⟨A, 𝐶⟩} = {𝐶})
5 rnsnopg 4742 . . . . . . 7 (B 𝐹 → ran {⟨B, 𝐷⟩} = {𝐷})
65adantl 262 . . . . . 6 ((A 𝐸 B 𝐹) → ran {⟨B, 𝐷⟩} = {𝐷})
763ad2ant1 924 . . . . 5 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → ran {⟨B, 𝐷⟩} = {𝐷})
84, 7uneq12d 3092 . . . 4 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → (ran {⟨A, 𝐶⟩} ∪ ran {⟨B, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
9 df-pr 3374 . . . . . 6 {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = ({⟨A, 𝐶⟩} ∪ {⟨B, 𝐷⟩})
109rneqi 4505 . . . . 5 ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = ran ({⟨A, 𝐶⟩} ∪ {⟨B, 𝐷⟩})
11 rnun 4675 . . . . 5 ran ({⟨A, 𝐶⟩} ∪ {⟨B, 𝐷⟩}) = (ran {⟨A, 𝐶⟩} ∪ ran {⟨B, 𝐷⟩})
1210, 11eqtri 2057 . . . 4 ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = (ran {⟨A, 𝐶⟩} ∪ ran {⟨B, 𝐷⟩})
13 df-pr 3374 . . . 4 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
148, 12, 133eqtr4g 2094 . . 3 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = {𝐶, 𝐷})
15 eqimss 2991 . . 3 (ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = {𝐶, 𝐷} → ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} ⊆ {𝐶, 𝐷})
1614, 15syl 14 . 2 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} ⊆ {𝐶, 𝐷})
17 df-f 4849 . 2 ({⟨A, 𝐶⟩, ⟨B, 𝐷⟩}:{A, B}⟶{𝐶, 𝐷} ↔ ({⟨A, 𝐶⟩, ⟨B, 𝐷⟩} Fn {A, B} ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} ⊆ {𝐶, 𝐷}))
181, 16, 17sylanbrc 394 1 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → {⟨A, 𝐶⟩, ⟨B, 𝐷⟩}:{A, B}⟶{𝐶, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884   = wceq 1242   wcel 1390  wne 2201  cun 2909  wss 2911  {csn 3367  {cpr 3368  cop 3370  ran crn 4289   Fn wfn 4840  wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by:  ftpg  5290
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