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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 4760 . . . . 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 4761 . . . . 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 6736 . . . . 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 1129 . 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 1191 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 Fn {𝐴, 𝐵})
16 simpr3 1197 . . . 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 1195 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → (𝐹𝐴) ∈ 𝐶)
1917, 18opelxpd 5724 . . . . 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 1196 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → (𝐹𝐵) ∈ 𝐶)
2220, 21opelxpd 5724 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → ⟨𝐵, (𝐹𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
2319, 22prssd 4822 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ⊆ ({𝐴, 𝐵} × 𝐶))
2416, 23eqsstrd 4018 . . 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 1087   = wceq 1540  wcel 2108  wss 3951  {cpr 4628  cop 4632   × cxp 5683   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569
This theorem is referenced by:  f1prex  7304  uhgrwkspthlem2  29774  rrx2xpref1o  48639
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