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Theorem f1ocnvfv3 5421
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-ontoB 𝐶 B) → (𝐹𝐶) = (x A (𝐹x) = 𝐶))
Distinct variable groups:   x,A   x,B   x,𝐶   x,𝐹

Proof of Theorem f1ocnvfv3
StepHypRef Expression
1 f1ocnvdm 5342 . . 3 ((𝐹:A1-1-ontoB 𝐶 B) → (𝐹𝐶) A)
2 f1ocnvfvb 5341 . . . . . 6 ((𝐹:A1-1-ontoB x A 𝐶 B) → ((𝐹x) = 𝐶 ↔ (𝐹𝐶) = x))
323expa 1088 . . . . 5 (((𝐹:A1-1-ontoB x A) 𝐶 B) → ((𝐹x) = 𝐶 ↔ (𝐹𝐶) = x))
43an32s 490 . . . 4 (((𝐹:A1-1-ontoB 𝐶 B) x A) → ((𝐹x) = 𝐶 ↔ (𝐹𝐶) = x))
5 eqcom 2020 . . . 4 (x = (𝐹𝐶) ↔ (𝐹𝐶) = x)
64, 5syl6bbr 187 . . 3 (((𝐹:A1-1-ontoB 𝐶 B) x A) → ((𝐹x) = 𝐶x = (𝐹𝐶)))
71, 6riota5 5413 . 2 ((𝐹:A1-1-ontoB 𝐶 B) → (x A (𝐹x) = 𝐶) = (𝐹𝐶))
87eqcomd 2023 1 ((𝐹:A1-1-ontoB 𝐶 B) → (𝐹𝐶) = (x A (𝐹x) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226   wcel 1370  ccnv 4267  1-1-ontowf1o 4824  cfv 4825  crio 5388
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-reu 2287  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-f1 4830  df-fo 4831  df-f1o 4832  df-fv 4833  df-riota 5389
This theorem is referenced by: (None)
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