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Theorem fpr2g 7162
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 486 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹:{𝐴, 𝐵}⟶𝐶)
2 prid1g 4722 . . . . 5 (𝐴𝑉𝐴 ∈ {𝐴, 𝐵})
32ad2antrr 725 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐴 ∈ {𝐴, 𝐵})
41, 3ffvelcdmd 7037 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹𝐴) ∈ 𝐶)
5 prid2g 4723 . . . . 5 (𝐵𝑊𝐵 ∈ {𝐴, 𝐵})
65ad2antlr 726 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐵 ∈ {𝐴, 𝐵})
71, 6ffvelcdmd 7037 . . 3 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹𝐵) ∈ 𝐶)
8 ffn 6669 . . . . 5 (𝐹:{𝐴, 𝐵}⟶𝐶𝐹 Fn {𝐴, 𝐵})
98adantl 483 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → 𝐹 Fn {𝐴, 𝐵})
10 fnpr2g 7161 . . . . 5 ((𝐴𝑉𝐵𝑊) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
1110adantr 482 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ 𝐹:{𝐴, 𝐵}⟶𝐶) → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩}))
129, 11mpbid 231 . . 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 479 . . . 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 725 . . . . . 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 5672 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → ⟨𝐴, (𝐹𝐴)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
205ad2antlr 726 . . . . . 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 5672 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → ⟨𝐵, (𝐹𝐵)⟩ ∈ ({𝐴, 𝐵} × 𝐶))
2319, 22prssd 4783 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩} ⊆ ({𝐴, 𝐵} × 𝐶))
2416, 23eqsstrd 3983 . . 3 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶))
25 dff2 7050 . . 3 (𝐹:{𝐴, 𝐵}⟶𝐶 ↔ (𝐹 Fn {𝐴, 𝐵} ∧ 𝐹 ⊆ ({𝐴, 𝐵} × 𝐶)))
2615, 24, 25sylanbrc 584 . 2 (((𝐴𝑉𝐵𝑊) ∧ ((𝐹𝐴) ∈ 𝐶 ∧ (𝐹𝐵) ∈ 𝐶𝐹 = {⟨𝐴, (𝐹𝐴)⟩, ⟨𝐵, (𝐹𝐵)⟩})) → 𝐹:{𝐴, 𝐵}⟶𝐶)
2713, 26impbida 800 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wss 3911  {cpr 4589  cop 4593   × cxp 5632   Fn wfn 6492  wf 6493  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505
This theorem is referenced by:  f1prex  7231  uhgrwkspthlem2  28744  rrx2xpref1o  46890
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