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Theorem fprb 7125
Description: A condition for functionhood over a pair. (Contributed by Scott Fenton, 16-Sep-2013.)
Hypotheses
Ref Expression
fprb.1 𝐴 ∈ V
fprb.2 𝐵 ∈ V
Assertion
Ref Expression
fprb (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦

Proof of Theorem fprb
StepHypRef Expression
1 fprb.1 . . . . . . 7 𝐴 ∈ V
21prid1 4710 . . . . . 6 𝐴 ∈ {𝐴, 𝐵}
3 ffvelcdm 7015 . . . . . 6 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴 ∈ {𝐴, 𝐵}) → (𝐹𝐴) ∈ 𝑅)
42, 3mpan2 688 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝑅 → (𝐹𝐴) ∈ 𝑅)
54adantr 481 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹𝐴) ∈ 𝑅)
6 fprb.2 . . . . . . 7 𝐵 ∈ V
76prid2 4711 . . . . . 6 𝐵 ∈ {𝐴, 𝐵}
8 ffvelcdm 7015 . . . . . 6 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐵 ∈ {𝐴, 𝐵}) → (𝐹𝐵) ∈ 𝑅)
97, 8mpan2 688 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝑅 → (𝐹𝐵) ∈ 𝑅)
109adantr 481 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹𝐵) ∈ 𝑅)
11 fvex 6838 . . . . . . . 8 (𝐹𝐴) ∈ V
121, 11fvpr1 7121 . . . . . . 7 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
13 fvex 6838 . . . . . . . 8 (𝐹𝐵) ∈ V
146, 13fvpr2 7123 . . . . . . 7 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
15 fveq2 6825 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
16 fveq2 6825 . . . . . . . . . 10 (𝑥 = 𝐴 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
1715, 16eqeq12d 2752 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴)))
18 eqcom 2743 . . . . . . . . 9 ((𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
1917, 18bitrdi 286 . . . . . . . 8 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴)))
20 fveq2 6825 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
21 fveq2 6825 . . . . . . . . . 10 (𝑥 = 𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
2220, 21eqeq12d 2752 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵)))
23 eqcom 2743 . . . . . . . . 9 ((𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
2422, 23bitrdi 286 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵)))
251, 6, 19, 24ralpr 4648 . . . . . . 7 (∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴) ∧ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵)))
2612, 14, 25sylanbrc 583 . . . . . 6 (𝐴𝐵 → ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
2726adantl 482 . . . . 5 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
28 ffn 6651 . . . . . 6 (𝐹:{𝐴, 𝐵}⟶𝑅𝐹 Fn {𝐴, 𝐵})
291, 6, 11, 13fpr 7082 . . . . . . 7 (𝐴𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}:{𝐴, 𝐵}⟶{(𝐹𝐴), (𝐹𝐵)})
3029ffnd 6652 . . . . . 6 (𝐴𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵})
31 eqfnfv 6965 . . . . . 6 ((𝐹 Fn {𝐴, 𝐵} ∧ {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵}) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
3228, 30, 31syl2an 596 . . . . 5 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
3327, 32mpbird 256 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
34 opeq2 4818 . . . . . . 7 (𝑥 = (𝐹𝐴) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝐹𝐴)⟩)
3534preq1d 4687 . . . . . 6 (𝑥 = (𝐹𝐴) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩})
3635eqeq2d 2747 . . . . 5 (𝑥 = (𝐹𝐴) → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩}))
37 opeq2 4818 . . . . . . 7 (𝑦 = (𝐹𝐵) → ⟨𝐵, 𝑦⟩ = ⟨𝐵, (𝐹𝐵)⟩)
3837preq2d 4688 . . . . . 6 (𝑦 = (𝐹𝐵) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
3938eqeq2d 2747 . . . . 5 (𝑦 = (𝐹𝐵) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
4036, 39rspc2ev 3581 . . . 4 (((𝐹𝐴) ∈ 𝑅 ∧ (𝐹𝐵) ∈ 𝑅𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩})
415, 10, 33, 40syl3anc 1370 . . 3 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩})
4241expcom 414 . 2 (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 → ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))
43 vex 3445 . . . . . . 7 𝑥 ∈ V
44 vex 3445 . . . . . . 7 𝑦 ∈ V
451, 6, 43, 44fpr 7082 . . . . . 6 (𝐴𝐵 → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶{𝑥, 𝑦})
46 prssi 4768 . . . . . 6 ((𝑥𝑅𝑦𝑅) → {𝑥, 𝑦} ⊆ 𝑅)
47 fss 6668 . . . . . 6 (({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑅) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅)
4845, 46, 47syl2an 596 . . . . 5 ((𝐴𝐵 ∧ (𝑥𝑅𝑦𝑅)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅)
4948ex 413 . . . 4 (𝐴𝐵 → ((𝑥𝑅𝑦𝑅) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅))
50 feq1 6632 . . . . 5 (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅))
5150biimprcd 249 . . . 4 ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅 → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅))
5249, 51syl6 35 . . 3 (𝐴𝐵 → ((𝑥𝑅𝑦𝑅) → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅)))
5352rexlimdvv 3200 . 2 (𝐴𝐵 → (∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅))
5442, 53impbid 211 1 (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1540  wcel 2105  wne 2940  wral 3061  wrex 3070  Vcvv 3441  wss 3898  {cpr 4575  cop 4579   Fn wfn 6474  wf 6475  cfv 6479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5243  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-fv 6487
This theorem is referenced by:  2arymaptf1  46359  prelrrx2b  46420
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