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Theorem f1ocnvfv3 7353
Description: Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
f1ocnvfv3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝑥𝐴 (𝐹𝑥) = 𝐶))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹

Proof of Theorem f1ocnvfv3
StepHypRef Expression
1 f1ocnvdm 7232 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) ∈ 𝐴)
2 f1ocnvfvb 7226 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝑥𝐴𝐶𝐵) → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝐶) = 𝑥))
323expa 1119 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝑥𝐴) ∧ 𝐶𝐵) → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝐶) = 𝑥))
43an32s 651 . . . 4 (((𝐹:𝐴1-1-onto𝐵𝐶𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝐶) = 𝑥))
5 eqcom 2744 . . . 4 (𝑥 = (𝐹𝐶) ↔ (𝐹𝐶) = 𝑥)
64, 5bitr4di 289 . . 3 (((𝐹:𝐴1-1-onto𝐵𝐶𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 𝐶𝑥 = (𝐹𝐶)))
71, 6riota5 7344 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝑥𝐴 (𝐹𝑥) = 𝐶) = (𝐹𝐶))
87eqcomd 2743 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝑥𝐴 (𝐹𝑥) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  ccnv 5633  1-1-ontowf1o 6496  cfv 6497  crio 7313
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 2708  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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  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-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  df-riota 7314
This theorem is referenced by: (None)
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