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Theorem fpr2g 7231
Description: A function that maps a pair to a class is a pair of ordered pairs. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Assertion
Ref Expression
fpr2g ((𝐴𝑉𝐵𝑊) → (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})))

Proof of Theorem fpr2g
StepHypRef Expression
1 simpr 484 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹:{𝐴, 𝐵}⟶𝐶)
2 prid1g 4765 . . . . 5 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
32ad2antrr 726 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐴 ∈ {𝐴, 𝐵})
41, 3ffvelcdmd 7105 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹𝐴) ∈ 𝐶)
5 prid2g 4766 . . . . 5 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
65ad2antlr 727 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐵 ∈ {𝐴, 𝐵})
71, 6ffvelcdmd 7105 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹𝐵) ∈ 𝐶)
8 ffn 6737 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝐶𝐹 Fn {𝐴, 𝐵})
98adantl 481 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹 Fn {𝐴, 𝐵})
10 fnpr2g 7230 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1110adantr 480 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
129, 11mpbid 232 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
134, 7, 123jca 1127 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1410biimpar 477 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → 𝐹 Fn {𝐴, 𝐵})
15143ad2antr3 1189 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 Fn {𝐴, 𝐵})
16 simpr3 1195 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
172ad2antrr 726 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐴 ∈ {𝐴, 𝐵})
18 simpr1 1193 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → (𝐹𝐴) ∈ 𝐶)
1917, 18opelxpd 5728 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → ⟨𝐴, (𝐹𝐴)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
205ad2antlr 727 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐵 ∈ {𝐴, 𝐵})
21 simpr2 1194 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → (𝐹𝐵) ∈ 𝐶)
2220, 21opelxpd 5728 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → ⟨𝐵, (𝐹𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
2319, 22prssd 4827 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ⊆ ({𝐴, 𝐵} × 𝐶))
2416, 23eqsstrd 4034 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶))
25 dff2 7119 . . 3 (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ (𝐹 Fn {𝐴, 𝐵} ∧ 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶)))
2615, 24, 25sylanbrc 583 . 2 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹:{𝐴, 𝐵}⟶𝐶)
2713, 26impbida 801 1 ((𝐴𝑉𝐵𝑊) → (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963  {cpr 4633  cop 4637   × cxp 5687   Fn wfn 6558  wf 6559  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by:  f1prex  7304  uhgrwkspthlem2  29787  rrx2xpref1o  48568
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