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Theorem f1ompt 5347
 Description: Express bijection for a mapping operation. (Contributed by Mario Carneiro, 30-May-2015.) (Revised by Mario Carneiro, 4-Dec-2016.)
Hypothesis
Ref Expression
fmpt.1 𝐹 = (𝑥𝐴𝐶)
Assertion
Ref Expression
f1ompt (𝐹:𝐴1-1-onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑦,𝐹
Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem f1ompt
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ffn 5073 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dff1o4 5161 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
32baib 839 . . . . 5 (𝐹 Fn 𝐴 → (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐵))
41, 3syl 14 . . . 4 (𝐹:𝐴𝐵 → (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐵))
5 fnres 5042 . . . . . 6 ((𝐹𝐵) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑧 𝑦𝐹𝑧)
6 nfcv 2194 . . . . . . . . . 10 𝑥𝑧
7 fmpt.1 . . . . . . . . . . 11 𝐹 = (𝑥𝐴𝐶)
8 nfmpt1 3877 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐶)
97, 8nfcxfr 2191 . . . . . . . . . 10 𝑥𝐹
10 nfcv 2194 . . . . . . . . . 10 𝑥𝑦
116, 9, 10nfbr 3835 . . . . . . . . 9 𝑥 𝑧𝐹𝑦
12 nfv 1437 . . . . . . . . 9 𝑧(𝑥𝐴𝑦 = 𝐶)
13 breq1 3794 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝐹𝑦𝑥𝐹𝑦))
14 df-mpt 3847 . . . . . . . . . . . . 13 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
157, 14eqtri 2076 . . . . . . . . . . . 12 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1615breqi 3797 . . . . . . . . . . 11 (𝑥𝐹𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦)
17 df-br 3792 . . . . . . . . . . . 12 (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
18 opabid 4021 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ↔ (𝑥𝐴𝑦 = 𝐶))
1917, 18bitri 177 . . . . . . . . . . 11 (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦 ↔ (𝑥𝐴𝑦 = 𝐶))
2016, 19bitri 177 . . . . . . . . . 10 (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐶))
2113, 20syl6bb 189 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐶)))
2211, 12, 21cbveu 1940 . . . . . . . 8 (∃!𝑧 𝑧𝐹𝑦 ↔ ∃!𝑥(𝑥𝐴𝑦 = 𝐶))
23 vex 2577 . . . . . . . . . 10 𝑦 ∈ V
24 vex 2577 . . . . . . . . . 10 𝑧 ∈ V
2523, 24brcnv 4545 . . . . . . . . 9 (𝑦𝐹𝑧𝑧𝐹𝑦)
2625eubii 1925 . . . . . . . 8 (∃!𝑧 𝑦𝐹𝑧 ↔ ∃!𝑧 𝑧𝐹𝑦)
27 df-reu 2330 . . . . . . . 8 (∃!𝑥𝐴 𝑦 = 𝐶 ↔ ∃!𝑥(𝑥𝐴𝑦 = 𝐶))
2822, 26, 273bitr4i 205 . . . . . . 7 (∃!𝑧 𝑦𝐹𝑧 ↔ ∃!𝑥𝐴 𝑦 = 𝐶)
2928ralbii 2347 . . . . . 6 (∀𝑦𝐵 ∃!𝑧 𝑦𝐹𝑧 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶)
305, 29bitri 177 . . . . 5 ((𝐹𝐵) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶)
31 relcnv 4730 . . . . . . 7 Rel 𝐹
32 df-rn 4383 . . . . . . . 8 ran 𝐹 = dom 𝐹
33 frn 5079 . . . . . . . 8 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
3432, 33syl5eqssr 3017 . . . . . . 7 (𝐹:𝐴𝐵 → dom 𝐹𝐵)
35 relssres 4675 . . . . . . 7 ((Rel 𝐹 ∧ dom 𝐹𝐵) → (𝐹𝐵) = 𝐹)
3631, 34, 35sylancr 399 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐹)
3736fneq1d 5016 . . . . 5 (𝐹:𝐴𝐵 → ((𝐹𝐵) Fn 𝐵𝐹 Fn 𝐵))
3830, 37syl5bbr 187 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶𝐹 Fn 𝐵))
394, 38bitr4d 184 . . 3 (𝐹:𝐴𝐵 → (𝐹:𝐴1-1-onto𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
4039pm5.32i 435 . 2 ((𝐹:𝐴𝐵𝐹:𝐴1-1-onto𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
41 f1of 5153 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
4241pm4.71ri 378 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴𝐵𝐹:𝐴1-1-onto𝐵))
437fmpt 5346 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
4443anbi1i 439 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
4540, 42, 443bitr4i 205 1 (𝐹:𝐴1-1-onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   ↔ wb 102   = wceq 1259   ∈ wcel 1409  ∃!weu 1916  ∀wral 2323  ∃!wreu 2325   ⊆ wss 2944  ⟨cop 3405   class class class wbr 3791  {copab 3844   ↦ cmpt 3845  ◡ccnv 4371  dom cdm 4372  ran crn 4373   ↾ cres 4374  Rel wrel 4377   Fn wfn 4924  ⟶wf 4925  –1-1-onto→wf1o 4928 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937 This theorem is referenced by:  icoshftf1o  8959
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