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Theorem f1ocnvfv3 6792
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 6686 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) ∈ 𝐴)
2 f1ocnvfvb 6681 . . . . . 6 ((𝐹:𝐴1-1-onto𝐵𝑥𝐴𝐶𝐵) → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝐶) = 𝑥))
323expa 1111 . . . . 5 (((𝐹:𝐴1-1-onto𝐵𝑥𝐴) ∧ 𝐶𝐵) → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝐶) = 𝑥))
43an32s 631 . . . 4 (((𝐹:𝐴1-1-onto𝐵𝐶𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 𝐶 ↔ (𝐹𝐶) = 𝑥))
5 eqcom 2778 . . . 4 (𝑥 = (𝐹𝐶) ↔ (𝐹𝐶) = 𝑥)
64, 5syl6bbr 278 . . 3 (((𝐹:𝐴1-1-onto𝐵𝐶𝐵) ∧ 𝑥𝐴) → ((𝐹𝑥) = 𝐶𝑥 = (𝐹𝐶)))
71, 6riota5 6783 . 2 ((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝑥𝐴 (𝐹𝑥) = 𝐶) = (𝐹𝐶))
87eqcomd 2777 1 ((𝐹:𝐴1-1-onto𝐵𝐶𝐵) → (𝐹𝐶) = (𝑥𝐴 (𝐹𝑥) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1631  wcel 2145  ccnv 5249  1-1-ontowf1o 6029  cfv 6030  crio 6756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6757
This theorem is referenced by: (None)
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