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Theorem fnprb 6377
 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.) Revised to eliminate unnecessary antecedent 𝐴 ≠ 𝐵. (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 6337 . . . . 5 (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
3 dfsn2 4138 . . . . . 6 {𝐴} = {𝐴, 𝐴}
43fneq2i 5900 . . . . 5 (𝐹 Fn {𝐴} ↔ 𝐹 Fn {𝐴, 𝐴})
5 dfsn2 4138 . . . . . 6 {⟨𝐴, (𝐹𝐴)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩}
65eqeq2i 2622 . . . . 5 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩})
72, 4, 63bitr3i 289 . . . 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 4213 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
109fneq2d 5896 . . 3 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴} ↔ 𝐹 Fn {𝐴, 𝐵}))
11 id 22 . . . . . 6 (𝐴 = 𝐵𝐴 = 𝐵)
12 fveq2 6103 . . . . . 6 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
1311, 12opeq12d 4348 . . . . 5 (𝐴 = 𝐵 → ⟨𝐴, (𝐹𝐴)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
1413preq2d 4219 . . . 4 (𝐴 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1514eqeq2d 2620 . . 3 (𝐴 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
168, 10, 153bitr3d 297 . 2 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
17 fndm 5904 . . . . . 6 (𝐹 Fn {𝐴, 𝐵} → dom 𝐹 = {𝐴, 𝐵})
18 fvex 6113 . . . . . . 7 (𝐹𝐴) ∈ V
19 fvex 6113 . . . . . . 7 (𝐹𝐵) ∈ V
2018, 19dmprop 5528 . . . . . 6 dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {𝐴, 𝐵}
2117, 20syl6eqr 2662 . . . . 5 (𝐹 Fn {𝐴, 𝐵} → dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
2221adantl 481 . . . 4 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
2317adantl 481 . . . . . . 7 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → dom 𝐹 = {𝐴, 𝐵})
2423eleq2d 2673 . . . . . 6 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ dom 𝐹𝑥 ∈ {𝐴, 𝐵}))
25 vex 3176 . . . . . . . 8 𝑥 ∈ V
2625elpr 4146 . . . . . . 7 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
271, 18fvpr1 6361 . . . . . . . . . . 11 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
2827adantr 480 . . . . . . . . . 10 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
2928eqcomd 2616 . . . . . . . . 9 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
30 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
31 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐴 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
3230, 31eqeq12d 2625 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴)))
3329, 32syl5ibrcom 236 . . . . . . . 8 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 = 𝐴 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
34 fnprb.b . . . . . . . . . . . 12 𝐵 ∈ V
3534, 19fvpr2 6362 . . . . . . . . . . 11 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
3635adantr 480 . . . . . . . . . 10 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
3736eqcomd 2616 . . . . . . . . 9 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
38 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
39 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
4038, 39eqeq12d 2625 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵)))
4137, 40syl5ibrcom 236 . . . . . . . 8 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 = 𝐵 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4233, 41jaod 394 . . . . . . 7 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ((𝑥 = 𝐴𝑥 = 𝐵) → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4326, 42syl5bi 231 . . . . . 6 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ {𝐴, 𝐵} → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4424, 43sylbid 229 . . . . 5 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4544ralrimiv 2948 . . . 4 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
46 fnfun 5902 . . . . 5 (𝐹 Fn {𝐴, 𝐵} → Fun 𝐹)
471, 34, 18, 19funpr 5858 . . . . 5 (𝐴𝐵 → Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
48 eqfunfv 6224 . . . . 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 494 . . . 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 959 . . 3 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
5120a1i 11 . . . . 5 (𝐴𝐵 → dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {𝐴, 𝐵})
52 df-fn 5807 . . . . 5 ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵} ↔ (Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ∧ dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {𝐴, 𝐵}))
5347, 51, 52sylanbrc 695 . . . 4 (𝐴𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵})
54 fneq1 5893 . . . . 5 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} → (𝐹 Fn {𝐴, 𝐵} ↔ {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵}))
5554biimprd 237 . . . 4 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵} → 𝐹 Fn {𝐴, 𝐵}))
5653, 55mpan9 485 . . 3 ((𝐴𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → 𝐹 Fn {𝐴, 𝐵})
5750, 56impbida 873 . 2 (𝐴𝐵 → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
5816, 57pm2.61ine 2865 1 (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173  {csn 4125  {cpr 4127  ⟨cop 4131  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812 This theorem is referenced by:  fntpb  6378  fnpr2g  6379  wrd2pr2op  13535
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