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Theorem fprb 31997
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 4441 . . . . . 6 𝐴 ∈ {𝐴, 𝐵}
3 ffvelrn 6521 . . . . . 6 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴 ∈ {𝐴, 𝐵}) → (𝐹𝐴) ∈ 𝑅)
42, 3mpan2 709 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝑅 → (𝐹𝐴) ∈ 𝑅)
54adantr 472 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹𝐴) ∈ 𝑅)
6 fprb.2 . . . . . . 7 𝐵 ∈ V
76prid2 4442 . . . . . 6 𝐵 ∈ {𝐴, 𝐵}
8 ffvelrn 6521 . . . . . 6 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐵 ∈ {𝐴, 𝐵}) → (𝐹𝐵) ∈ 𝑅)
97, 8mpan2 709 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝑅 → (𝐹𝐵) ∈ 𝑅)
109adantr 472 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹𝐵) ∈ 𝑅)
11 fvex 6363 . . . . . . . 8 (𝐹𝐴) ∈ V
121, 11fvpr1 6621 . . . . . . 7 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
13 fvex 6363 . . . . . . . 8 (𝐹𝐵) ∈ V
146, 13fvpr2 6622 . . . . . . 7 (𝐴𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
15 fveq2 6353 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
16 fveq2 6353 . . . . . . . . . 10 (𝑥 = 𝐴 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴))
1715, 16eqeq12d 2775 . . . . . . . . 9 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴)))
18 eqcom 2767 . . . . . . . . 9 ((𝐹𝐴) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴))
1917, 18syl6bb 276 . . . . . . . 8 (𝑥 = 𝐴 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐴) = (𝐹𝐴)))
20 fveq2 6353 . . . . . . . . . 10 (𝑥 = 𝐵 → (𝐹𝑥) = (𝐹𝐵))
21 fveq2 6353 . . . . . . . . . 10 (𝑥 = 𝐵 → ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵))
2220, 21eqeq12d 2775 . . . . . . . . 9 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ (𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵)))
23 eqcom 2767 . . . . . . . . 9 ((𝐹𝐵) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵))
2422, 23syl6bb 276 . . . . . . . 8 (𝑥 = 𝐵 → ((𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥) ↔ ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝐵) = (𝐹𝐵)))
251, 6, 19, 24ralpr 4382 . . . . . . 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 701 . . . . . 6 (𝐴𝐵 → ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
2726adantl 473 . . . . 5 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥))
28 ffn 6206 . . . . . 6 (𝐹:{𝐴, 𝐵}⟶𝑅𝐹 Fn {𝐴, 𝐵})
291, 6, 11, 13fpr 6585 . . . . . . 7 (𝐴𝐵 → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}:{𝐴, 𝐵}⟶{(𝐹𝐴), (𝐹𝐵)})
30 ffn 6206 . . . . . . 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 6475 . . . . . 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 495 . . . . 5 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ↔ ∀𝑥 ∈ {𝐴, 𝐵} (𝐹𝑥) = ({⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}‘𝑥)))
3427, 33mpbird 247 . . . 4 ((𝐹:{𝐴, 𝐵}⟶𝑅𝐴𝐵) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
35 opeq2 4554 . . . . . . 7 (𝑥 = (𝐹𝐴) → ⟨𝐴, 𝑥⟩ = ⟨𝐴, (𝐹𝐴)⟩)
3635preq1d 4418 . . . . . 6 (𝑥 = (𝐹𝐴) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩})
3736eqeq2d 2770 . . . . 5 (𝑥 = (𝐹𝐴) → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩}))
38 opeq2 4554 . . . . . . 7 (𝑦 = (𝐹𝐵) → ⟨𝐵, 𝑦⟩ = ⟨𝐵, (𝐹𝐵)⟩)
3938preq2d 4419 . . . . . 6 (𝑦 = (𝐹𝐵) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩} = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
4039eqeq2d 2770 . . . . 5 (𝑦 = (𝐹𝐵) → (𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, 𝑦⟩} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
4137, 40rspc2ev 3463 . . . 4 (((𝐹𝐴) ∈ 𝑅 ∧ (𝐹𝐵) ∈ 𝑅𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩})
425, 10, 34, 41syl3anc 1477 . . 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 3343 . . . . . . 7 𝑥 ∈ V
45 vex 3343 . . . . . . 7 𝑦 ∈ V
461, 6, 44, 45fpr 6585 . . . . . 6 (𝐴𝐵 → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶{𝑥, 𝑦})
47 prssi 4498 . . . . . 6 ((𝑥𝑅𝑦𝑅) → {𝑥, 𝑦} ⊆ 𝑅)
48 fss 6217 . . . . . 6 (({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶{𝑥, 𝑦} ∧ {𝑥, 𝑦} ⊆ 𝑅) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅)
4946, 47, 48syl2an 495 . . . . 5 ((𝐴𝐵 ∧ (𝑥𝑅𝑦𝑅)) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅)
5049ex 449 . . . 4 (𝐴𝐵 → ((𝑥𝑅𝑦𝑅) → {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅))
51 feq1 6187 . . . . 5 (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅))
5251biimprcd 240 . . . 4 ({⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}:{𝐴, 𝐵}⟶𝑅 → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅))
5350, 52syl6 35 . . 3 (𝐴𝐵 → ((𝑥𝑅𝑦𝑅) → (𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅)))
5453rexlimdvv 3175 . 2 (𝐴𝐵 → (∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩} → 𝐹:{𝐴, 𝐵}⟶𝑅))
5543, 54impbid 202 1 (𝐴𝐵 → (𝐹:{𝐴, 𝐵}⟶𝑅 ↔ ∃𝑥𝑅𝑦𝑅 𝐹 = {⟨𝐴, 𝑥⟩, ⟨𝐵, 𝑦⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  wss 3715  {cpr 4323  cop 4327   Fn wfn 6044  wf 6045  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057
This theorem is referenced by: (None)
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