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Theorem f1ompt 6338
 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 6002 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dff1o4 6102 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
32baib 943 . . . . 5 (𝐹 Fn 𝐴 → (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐵))
41, 3syl 17 . . . 4 (𝐹:𝐴𝐵 → (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐵))
5 fnres 5965 . . . . . 6 ((𝐹𝐵) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑧 𝑦𝐹𝑧)
6 nfcv 2761 . . . . . . . . . 10 𝑥𝑧
7 fmpt.1 . . . . . . . . . . 11 𝐹 = (𝑥𝐴𝐶)
8 nfmpt1 4707 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐶)
97, 8nfcxfr 2759 . . . . . . . . . 10 𝑥𝐹
10 nfcv 2761 . . . . . . . . . 10 𝑥𝑦
116, 9, 10nfbr 4659 . . . . . . . . 9 𝑥 𝑧𝐹𝑦
12 nfv 1840 . . . . . . . . 9 𝑧(𝑥𝐴𝑦 = 𝐶)
13 breq1 4616 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝐹𝑦𝑥𝐹𝑦))
14 df-mpt 4675 . . . . . . . . . . . . 13 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
157, 14eqtri 2643 . . . . . . . . . . . 12 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1615breqi 4619 . . . . . . . . . . 11 (𝑥𝐹𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦)
17 df-br 4614 . . . . . . . . . . . 12 (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
18 opabid 4942 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)} ↔ (𝑥𝐴𝑦 = 𝐶))
1917, 18bitri 264 . . . . . . . . . . 11 (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦 ↔ (𝑥𝐴𝑦 = 𝐶))
2016, 19bitri 264 . . . . . . . . . 10 (𝑥𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐶))
2113, 20syl6bb 276 . . . . . . . . 9 (𝑧 = 𝑥 → (𝑧𝐹𝑦 ↔ (𝑥𝐴𝑦 = 𝐶)))
2211, 12, 21cbveu 2504 . . . . . . . 8 (∃!𝑧 𝑧𝐹𝑦 ↔ ∃!𝑥(𝑥𝐴𝑦 = 𝐶))
23 vex 3189 . . . . . . . . . 10 𝑦 ∈ V
24 vex 3189 . . . . . . . . . 10 𝑧 ∈ V
2523, 24brcnv 5265 . . . . . . . . 9 (𝑦𝐹𝑧𝑧𝐹𝑦)
2625eubii 2491 . . . . . . . 8 (∃!𝑧 𝑦𝐹𝑧 ↔ ∃!𝑧 𝑧𝐹𝑦)
27 df-reu 2914 . . . . . . . 8 (∃!𝑥𝐴 𝑦 = 𝐶 ↔ ∃!𝑥(𝑥𝐴𝑦 = 𝐶))
2822, 26, 273bitr4i 292 . . . . . . 7 (∃!𝑧 𝑦𝐹𝑧 ↔ ∃!𝑥𝐴 𝑦 = 𝐶)
2928ralbii 2974 . . . . . 6 (∀𝑦𝐵 ∃!𝑧 𝑦𝐹𝑧 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶)
305, 29bitri 264 . . . . 5 ((𝐹𝐵) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶)
31 relcnv 5462 . . . . . . 7 Rel 𝐹
32 df-rn 5085 . . . . . . . 8 ran 𝐹 = dom 𝐹
33 frn 6010 . . . . . . . 8 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
3432, 33syl5eqssr 3629 . . . . . . 7 (𝐹:𝐴𝐵 → dom 𝐹𝐵)
35 relssres 5396 . . . . . . 7 ((Rel 𝐹 ∧ dom 𝐹𝐵) → (𝐹𝐵) = 𝐹)
3631, 34, 35sylancr 694 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐹)
3736fneq1d 5939 . . . . 5 (𝐹:𝐴𝐵 → ((𝐹𝐵) Fn 𝐵𝐹 Fn 𝐵))
3830, 37syl5bbr 274 . . . 4 (𝐹:𝐴𝐵 → (∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶𝐹 Fn 𝐵))
394, 38bitr4d 271 . . 3 (𝐹:𝐴𝐵 → (𝐹:𝐴1-1-onto𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
4039pm5.32i 668 . 2 ((𝐹:𝐴𝐵𝐹:𝐴1-1-onto𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
41 f1of 6094 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
4241pm4.71ri 664 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴𝐵𝐹:𝐴1-1-onto𝐵))
437fmpt 6337 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
4443anbi1i 730 . 2 ((∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
4540, 42, 443bitr4i 292 1 (𝐹:𝐴1-1-onto𝐵 ↔ (∀𝑥𝐴 𝐶𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃!weu 2469  ∀wral 2907  ∃!wreu 2909   ⊆ wss 3555  ⟨cop 4154   class class class wbr 4613  {copab 4672   ↦ cmpt 4673  ◡ccnv 5073  dom cdm 5074  ran crn 5075   ↾ cres 5076  Rel wrel 5079   Fn wfn 5842  ⟶wf 5843  –1-1-onto→wf1o 5846 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855 This theorem is referenced by:  oaf1o  7588  xpf1o  8066  icoshftf1o  12237  fprodser  14604  dfod2  17902  gsummptf1o  18283  nbusgrf1o0  26158  cusgrfilem2  26239  numclwlk2lem2f1o  27093  f1mptrn  29278  xrmulc1cn  29758  poimirlem4  33045  poimirlem16  33057  poimirlem17  33058  poimirlem19  33060  poimirlem20  33061
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