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Theorem fpr2g 6966
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 487 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹:{𝐴, 𝐵}⟶𝐶)
2 prid1g 4688 . . . . 5 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
32ad2antrr 724 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐴 ∈ {𝐴, 𝐵})
41, 3ffvelrnd 6845 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹𝐴) ∈ 𝐶)
5 prid2g 4689 . . . . 5 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
65ad2antlr 725 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐵 ∈ {𝐴, 𝐵})
71, 6ffvelrnd 6845 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹𝐵) ∈ 𝐶)
8 ffn 6507 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝐶𝐹 Fn {𝐴, 𝐵})
98adantl 484 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹 Fn {𝐴, 𝐵})
10 fnpr2g 6965 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1110adantr 483 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
129, 11mpbid 234 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
134, 7, 123jca 1122 . 2 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1410biimpar 480 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}) → 𝐹 Fn {𝐴, 𝐵})
15143ad2antr3 1184 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 Fn {𝐴, 𝐵})
16 simpr3 1190 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})
172ad2antrr 724 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐴 ∈ {𝐴, 𝐵})
18 simpr1 1188 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → (𝐹𝐴) ∈ 𝐶)
1917, 18opelxpd 5586 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → ⟨𝐴, (𝐹𝐴)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
205ad2antlr 725 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐵 ∈ {𝐴, 𝐵})
21 simpr2 1189 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → (𝐹𝐵) ∈ 𝐶)
2220, 21opelxpd 5586 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → ⟨𝐵, (𝐹𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
2319, 22prssd 4747 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ⊆ ({𝐴, 𝐵} × 𝐶))
2416, 23eqsstrd 4003 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶))
25 dff2 6858 . . 3 (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ (𝐹 Fn {𝐴, 𝐵} ∧ 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶)))
2615, 24, 25sylanbrc 585 . 2 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹:{𝐴, 𝐵}⟶𝐶)
2713, 26impbida 799 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 208  wa 398  w3a 1081   = wceq 1530  wcel 2107  wss 3934  {cpr 4561  cop 4565   × cxp 5546   Fn wfn 6343  wf 6344  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356
This theorem is referenced by:  f1prex  7032  uhgrwkspthlem2  27527  rrx2xpref1o  44695
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