Mathbox for Scott Fenton < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fprb Structured version   Visualization version   GIF version

Theorem fprb 30916
 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 4241 . . . . . 6 𝐴 ∈ {𝐴, 𝐵}
3 ffvelrn 6265 . . . . . 6 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴 ∈ {𝐴, 𝐵}) → (𝐹𝐴) ∈ 𝑅)
42, 3mpan2 703 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝑅 → (𝐹𝐴) ∈ 𝑅)
54adantr 480 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹𝐴) ∈ 𝑅)
6 fprb.2 . . . . . . 7 𝐵 ∈ V
76prid2 4242 . . . . . 6 𝐵 ∈ {𝐴, 𝐵}
8 ffvelrn 6265 . . . . . 6 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐵 ∈ {𝐴, 𝐵}) → (𝐹𝐵) ∈ 𝑅)
97, 8mpan2 703 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝑅 → (𝐹𝐵) ∈ 𝑅)
109adantr 480 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹𝐵) ∈ 𝑅)
11 fvex 6113 . . . . . . . 8 (𝐹𝐴) ∈ V
121, 11fvpr1 6361 . . . . . . 7 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
13 fvex 6113 . . . . . . . 8 (𝐹𝐵) ∈ V
146, 13fvpr2 6362 . . . . . . 7 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
15 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
16 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐴 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
1715, 16eqeq12d 2625 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴)))
18 eqcom 2617 . . . . . . . . 9 ((𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
1917, 18syl6bb 275 . . . . . . . 8 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴)))
20 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
21 fveq2 6103 . . . . . . . . . 10 (𝑥 = 𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
2220, 21eqeq12d 2625 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵)))
23 eqcom 2617 . . . . . . . . 9 ((𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
2422, 23syl6bb 275 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵)))
251, 6, 19, 24ralpr 4185 . . . . . . 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 695 . . . . . 6 (𝐴𝐵 → ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
2726adantl 481 . . . . 5 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
28 ffn 5958 . . . . . 6 (𝐹:{𝐴, 𝐵}⟶𝑅𝐹 Fn {𝐴, 𝐵})
291, 6, 11, 13fpr 6326 . . . . . . 7 (𝐴𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}:{𝐴, 𝐵}⟶{(𝐹𝐴), (𝐹𝐵)})
30 ffn 5958 . . . . . . 7 ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}:{𝐴, 𝐵}⟶{(𝐹𝐴), (𝐹𝐵)} → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵})
3129, 30syl 17 . . . . . 6 (𝐴𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} Fn {𝐴, 𝐵})
32 eqfnfv 6219 . . . . . 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)))
3328, 31, 32syl2an 493 . . . . 5 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
3427, 33mpbird 246 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
35 opeq2 4341 . . . . . . 7 (𝑥 = (𝐹𝐴) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝐹𝐴)⟩)
3635preq1d 4218 . . . . . 6 (𝑥 = (𝐹𝐴) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩})
3736eqeq2d 2620 . . . . 5 (𝑥 = (𝐹𝐴) → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩}))
38 opeq2 4341 . . . . . . 7 (𝑦 = (𝐹𝐵) → ⟨𝐵, 𝑦⟩ = ⟨𝐵, (𝐹𝐵)⟩)
3938preq2d 4219 . . . . . 6 (𝑦 = (𝐹𝐵) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
4039eqeq2d 2620 . . . . 5 (𝑦 = (𝐹𝐵) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
4137, 40rspc2ev 3295 . . . 4 (((𝐹𝐴) ∈ 𝑅 ∧ (𝐹𝐵) ∈ 𝑅𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩})
425, 10, 34, 41syl3anc 1318 . . 3 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩})
4342expcom 450 . 2 (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 → ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))
44 vex 3176 . . . . . . 7 𝑥 ∈ V
45 vex 3176 . . . . . . 7 𝑦 ∈ V
461, 6, 44, 45fpr 6326 . . . . . 6 (𝐴𝐵 → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶{𝑥, 𝑦})
47 prssi 4293 . . . . . 6 ((𝑥𝑅𝑦𝑅) → {𝑥, 𝑦} ⊆ 𝑅)
48 fss 5969 . . . . . 6 (({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑅) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅)
4946, 47, 48syl2an 493 . . . . 5 ((𝐴𝐵 ∧ (𝑥𝑅𝑦𝑅)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅)
5049ex 449 . . . 4 (𝐴𝐵 → ((𝑥𝑅𝑦𝑅) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅))
51 feq1 5939 . . . . 5 (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅))
5251biimprcd 239 . . . 4 ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅 → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅))
5350, 52syl6 34 . . 3 (𝐴𝐵 → ((𝑥𝑅𝑦𝑅) → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅)))
5453rexlimdvv 3019 . 2 (𝐴𝐵 → (∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅))
5543, 54impbid 201 1 (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  {cpr 4127  ⟨cop 4131   Fn wfn 5799  ⟶wf 5800  ‘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-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-fv 5812 This theorem is referenced by: (None)
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