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Theorem fprg 5271
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 4880 . 2 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} Fn {A, B})
2 rnsnopg 4726 . . . . . . 7 (A 𝐸 → ran {⟨A, 𝐶⟩} = {𝐶})
32adantr 261 . . . . . 6 ((A 𝐸 B 𝐹) → ran {⟨A, 𝐶⟩} = {𝐶})
433ad2ant1 913 . . . . 5 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → ran {⟨A, 𝐶⟩} = {𝐶})
5 rnsnopg 4726 . . . . . . 7 (B 𝐹 → ran {⟨B, 𝐷⟩} = {𝐷})
65adantl 262 . . . . . 6 ((A 𝐸 B 𝐹) → ran {⟨B, 𝐷⟩} = {𝐷})
763ad2ant1 913 . . . . 5 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → ran {⟨B, 𝐷⟩} = {𝐷})
84, 7uneq12d 3075 . . . 4 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → (ran {⟨A, 𝐶⟩} ∪ ran {⟨B, 𝐷⟩}) = ({𝐶} ∪ {𝐷}))
9 df-pr 3357 . . . . . 6 {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = ({⟨A, 𝐶⟩} ∪ {⟨B, 𝐷⟩})
109rneqi 4489 . . . . 5 ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = ran ({⟨A, 𝐶⟩} ∪ {⟨B, 𝐷⟩})
11 rnun 4659 . . . . 5 ran ({⟨A, 𝐶⟩} ∪ {⟨B, 𝐷⟩}) = (ran {⟨A, 𝐶⟩} ∪ ran {⟨B, 𝐷⟩})
1210, 11eqtri 2042 . . . 4 ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = (ran {⟨A, 𝐶⟩} ∪ ran {⟨B, 𝐷⟩})
13 df-pr 3357 . . . 4 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
148, 12, 133eqtr4g 2079 . . 3 (((A 𝐸 B 𝐹) (𝐶 𝐺 𝐷 𝐻) AB) → ran {⟨A, 𝐶⟩, ⟨B, 𝐷⟩} = {𝐶, 𝐷})
15 eqimss 2974 . . 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 4833 . 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 396 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 873   = wceq 1228   wcel 1374  wne 2186  cun 2892  wss 2894  {csn 3350  {cpr 3351  cop 3353  ran crn 4273   Fn wfn 4824  wf 4825
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-v 2537  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-fun 4831  df-fn 4832  df-f 4833
This theorem is referenced by:  ftpg  5272
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