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Theorem fpr2g 6987
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 488 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹:{𝐴, 𝐵}⟶𝐶)
2 prid1g 4652 . . . . 5 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
32ad2antrr 726 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐴 ∈ {𝐴, 𝐵})
41, 3ffvelrnd 6865 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹𝐴) ∈ 𝐶)
5 prid2g 4653 . . . . 5 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
65ad2antlr 727 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐵 ∈ {𝐴, 𝐵})
71, 6ffvelrnd 6865 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹𝐵) ∈ 𝐶)
8 ffn 6505 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝐶𝐹 Fn {𝐴, 𝐵})
98adantl 485 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹 Fn {𝐴, 𝐵})
10 fnpr2g 6986 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1110adantr 484 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
129, 11mpbid 235 . . 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 481 . . . 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 5564 . . . . 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 5564 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → ⟨𝐵, (𝐹𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
2319, 22prssd 4711 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ⊆ ({𝐴, 𝐵} × 𝐶))
2416, 23eqsstrd 3916 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶))
25 dff2 6878 . . 3 (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ (𝐹 Fn {𝐴, 𝐵} ∧ 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶)))
2615, 24, 25sylanbrc 586 . 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wss 3844  {cpr 4519  cop 4523   × cxp 5524   Fn wfn 6335  wf 6336  cfv 6340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348
This theorem is referenced by:  f1prex  7054  uhgrwkspthlem2  27698  rrx2xpref1o  45628
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