MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnprb Structured version   Visualization version   GIF version

Theorem fnprb 6952
Description: A function whose domain has at most two elements can be represented as a set of at most two ordered pairs. (Contributed by FL, 26-Jun-2011.) (Proof shortened by Scott Fenton, 12-Oct-2017.) Eliminate unnecessary antecedent 𝐴𝐵. (Revised by NM, 29-Dec-2018.)
Hypotheses
Ref Expression
fnprb.a 𝐴 ∈ V
fnprb.b 𝐵 ∈ V
Assertion
Ref Expression
fnprb (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})

Proof of Theorem fnprb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fnprb.a . . . . . 6 𝐴 ∈ V
21fnsnb 6909 . . . . 5 (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
3 dfsn2 4541 . . . . . 6 {𝐴} = {𝐴, 𝐴}
43fneq2i 6425 . . . . 5 (𝐹 Fn {𝐴} ↔ 𝐹 Fn {𝐴, 𝐴})
5 dfsn2 4541 . . . . . 6 {⟨𝐴, (𝐹𝐴)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩}
65eqeq2i 2814 . . . . 5 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩})
72, 4, 63bitr3i 304 . . . 4 (𝐹 Fn {𝐴, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩})
87a1i 11 . . 3 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩}))
9 preq2 4633 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
109fneq2d 6421 . . 3 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴} ↔ 𝐹 Fn {𝐴, 𝐵}))
11 id 22 . . . . . 6 (𝐴 = 𝐵𝐴 = 𝐵)
12 fveq2 6649 . . . . . 6 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
1311, 12opeq12d 4776 . . . . 5 (𝐴 = 𝐵 → ⟨𝐴, (𝐹𝐴)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
1413preq2d 4639 . . . 4 (𝐴 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1514eqeq2d 2812 . . 3 (𝐴 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
168, 10, 153bitr3d 312 . 2 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
17 fndm 6429 . . . . . 6 (𝐹 Fn {𝐴, 𝐵} → dom 𝐹 = {𝐴, 𝐵})
18 fvex 6662 . . . . . . 7 (𝐹𝐴) ∈ V
19 fvex 6662 . . . . . . 7 (𝐹𝐵) ∈ V
2018, 19dmprop 6045 . . . . . 6 dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {𝐴, 𝐵}
2117, 20eqtr4di 2854 . . . . 5 (𝐹 Fn {𝐴, 𝐵} → dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
2221adantl 485 . . . 4 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
2317adantl 485 . . . . . . 7 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → dom 𝐹 = {𝐴, 𝐵})
2423eleq2d 2878 . . . . . 6 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ dom 𝐹𝑥 ∈ {𝐴, 𝐵}))
25 vex 3447 . . . . . . . 8 𝑥 ∈ V
2625elpr 4551 . . . . . . 7 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
271, 18fvpr1 6933 . . . . . . . . . . 11 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
2827adantr 484 . . . . . . . . . 10 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
2928eqcomd 2807 . . . . . . . . 9 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
30 fveq2 6649 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
31 fveq2 6649 . . . . . . . . . 10 (𝑥 = 𝐴 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
3230, 31eqeq12d 2817 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴)))
3329, 32syl5ibrcom 250 . . . . . . . 8 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 = 𝐴 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
34 fnprb.b . . . . . . . . . . . 12 𝐵 ∈ V
3534, 19fvpr2 6934 . . . . . . . . . . 11 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
3635adantr 484 . . . . . . . . . 10 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
3736eqcomd 2807 . . . . . . . . 9 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
38 fveq2 6649 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
39 fveq2 6649 . . . . . . . . . 10 (𝑥 = 𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
4038, 39eqeq12d 2817 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵)))
4137, 40syl5ibrcom 250 . . . . . . . 8 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 = 𝐵 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4233, 41jaod 856 . . . . . . 7 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ((𝑥 = 𝐴𝑥 = 𝐵) → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4326, 42syl5bi 245 . . . . . 6 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ {𝐴, 𝐵} → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4424, 43sylbid 243 . . . . 5 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4544ralrimiv 3151 . . . 4 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
46 fnfun 6427 . . . . 5 (𝐹 Fn {𝐴, 𝐵} → Fun 𝐹)
471, 34, 18, 19funpr 6384 . . . . 5 (𝐴𝐵 → Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
48 eqfunfv 6788 . . . . 5 ((Fun 𝐹 ∧ Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))))
4946, 47, 48syl2anr 599 . . . 4 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))))
5022, 45, 49mpbir2and 712 . . 3 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
51 df-fn 6331 . . . . 5 ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵} ↔ (Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ∧ dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {𝐴, 𝐵}))
5247, 20, 51sylanblrc 593 . . . 4 (𝐴𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵})
53 fneq1 6418 . . . . 5 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} → (𝐹 Fn {𝐴, 𝐵} ↔ {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵}))
5453biimprd 251 . . . 4 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵} → 𝐹 Fn {𝐴, 𝐵}))
5552, 54mpan9 510 . . 3 ((𝐴𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → 𝐹 Fn {𝐴, 𝐵})
5650, 55impbida 800 . 2 (𝐴𝐵 → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
5716, 56pm2.61ine 3073 1 (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  wne 2990  wral 3109  Vcvv 3444  {csn 4528  {cpr 4530  cop 4534  dom cdm 5523  Fun wfun 6322   Fn wfn 6323  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336
This theorem is referenced by:  fntpb  6953  fnpr2g  6954  wrd2pr2op  14300
  Copyright terms: Public domain W3C validator