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Theorem fnprb 6697
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 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 6657 . . . . 5 (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩})
3 dfsn2 4383 . . . . . 6 {𝐴} = {𝐴, 𝐴}
43fneq2i 6197 . . . . 5 (𝐹 Fn {𝐴} ↔ 𝐹 Fn {𝐴, 𝐴})
5 dfsn2 4383 . . . . . 6 {⟨𝐴, (𝐹𝐴)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩}
65eqeq2i 2818 . . . . 5 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩})
72, 4, 63bitr3i 292 . . . 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 4460 . . . 4 (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵})
109fneq2d 6193 . . 3 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴} ↔ 𝐹 Fn {𝐴, 𝐵}))
11 id 22 . . . . . 6 (𝐴 = 𝐵𝐴 = 𝐵)
12 fveq2 6408 . . . . . 6 (𝐴 = 𝐵 → (𝐹𝐴) = (𝐹𝐵))
1311, 12opeq12d 4603 . . . . 5 (𝐴 = 𝐵 → ⟨𝐴, (𝐹𝐴)⟩ = ⟨𝐵, (𝐹𝐵)⟩)
1413preq2d 4466 . . . 4 (𝐴 = 𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
1514eqeq2d 2816 . . 3 (𝐴 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐴, (𝐹𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
168, 10, 153bitr3d 300 . 2 (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
17 fndm 6201 . . . . . 6 (𝐹 Fn {𝐴, 𝐵} → dom 𝐹 = {𝐴, 𝐵})
18 fvex 6421 . . . . . . 7 (𝐹𝐴) ∈ V
19 fvex 6421 . . . . . . 7 (𝐹𝐵) ∈ V
2018, 19dmprop 5822 . . . . . 6 dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {𝐴, 𝐵}
2117, 20syl6eqr 2858 . . . . 5 (𝐹 Fn {𝐴, 𝐵} → dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
2221adantl 469 . . . 4 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → dom 𝐹 = dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
2317adantl 469 . . . . . . 7 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → dom 𝐹 = {𝐴, 𝐵})
2423eleq2d 2871 . . . . . 6 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ dom 𝐹𝑥 ∈ {𝐴, 𝐵}))
25 vex 3394 . . . . . . . 8 𝑥 ∈ V
2625elpr 4393 . . . . . . 7 (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴𝑥 = 𝐵))
271, 18fvpr1 6681 . . . . . . . . . . 11 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
2827adantr 468 . . . . . . . . . 10 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
2928eqcomd 2812 . . . . . . . . 9 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
30 fveq2 6408 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
31 fveq2 6408 . . . . . . . . . 10 (𝑥 = 𝐴 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
3230, 31eqeq12d 2821 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴)))
3329, 32syl5ibrcom 238 . . . . . . . 8 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 = 𝐴 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
34 fnprb.b . . . . . . . . . . . 12 𝐵 ∈ V
3534, 19fvpr2 6682 . . . . . . . . . . 11 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
3635adantr 468 . . . . . . . . . 10 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
3736eqcomd 2812 . . . . . . . . 9 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
38 fveq2 6408 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
39 fveq2 6408 . . . . . . . . . 10 (𝑥 = 𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
4038, 39eqeq12d 2821 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵)))
4137, 40syl5ibrcom 238 . . . . . . . 8 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 = 𝐵 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4233, 41jaod 877 . . . . . . 7 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ((𝑥 = 𝐴𝑥 = 𝐵) → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4326, 42syl5bi 233 . . . . . 6 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ {𝐴, 𝐵} → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4424, 43sylbid 231 . . . . 5 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
4544ralrimiv 3153 . . . 4 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → ∀𝑥 ∈ dom 𝐹(𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
46 fnfun 6199 . . . . 5 (𝐹 Fn {𝐴, 𝐵} → Fun 𝐹)
471, 34, 18, 19funpr 6156 . . . . 5 (𝐴𝐵 → Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
48 eqfunfv 6538 . . . . 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 586 . . . 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 695 . . 3 ((𝐴𝐵𝐹 Fn {𝐴, 𝐵}) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
5120a1i 11 . . . . 5 (𝐴𝐵 → dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {𝐴, 𝐵})
52 df-fn 6104 . . . . 5 ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵} ↔ (Fun {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ∧ dom {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} = {𝐴, 𝐵}))
5347, 51, 52sylanbrc 574 . . . 4 (𝐴𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵})
54 fneq1 6190 . . . . 5 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} → (𝐹 Fn {𝐴, 𝐵} ↔ {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵}))
5554biimprd 239 . . . 4 (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵} → 𝐹 Fn {𝐴, 𝐵}))
5653, 55mpan9 498 . . 3 ((𝐴𝐵𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → 𝐹 Fn {𝐴, 𝐵})
5750, 56impbida 826 . 2 (𝐴𝐵 → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
5816, 57pm2.61ine 3061 1 (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  wo 865   = wceq 1637  wcel 2156  wne 2978  wral 3096  Vcvv 3391  {csn 4370  {cpr 4372  cop 4376  dom cdm 5311  Fun wfun 6095   Fn wfn 6096  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109
This theorem is referenced by:  fntpb  6698  fnpr2g  6699  wrd2pr2op  13912
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