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Theorem f1ompt 6348
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 6012 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dff1o4 6112 . . . . . 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 5975 . . . . . 6 ((𝐹𝐵) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑧 𝑦𝐹𝑧)
6 nfcv 2761 . . . . . . . . . 10 𝑥𝑧
7 fmpt.1 . . . . . . . . . . 11 𝐹 = (𝑥𝐴𝐶)
8 nfmpt1 4717 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐶)
97, 8nfcxfr 2759 . . . . . . . . . 10 𝑥𝐹
10 nfcv 2761 . . . . . . . . . 10 𝑥𝑦
116, 9, 10nfbr 4669 . . . . . . . . 9 𝑥 𝑧𝐹𝑦
12 nfv 1840 . . . . . . . . 9 𝑧(𝑥𝐴𝑦 = 𝐶)
13 breq1 4626 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝐹𝑦𝑥𝐹𝑦))
14 df-mpt 4685 . . . . . . . . . . . . 13 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
157, 14eqtri 2643 . . . . . . . . . . . 12 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1615breqi 4629 . . . . . . . . . . 11 (𝑥𝐹𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦)
17 df-br 4624 . . . . . . . . . . . 12 (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
18 opabid 4952 . . . . . . . . . . . 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 3193 . . . . . . . . . 10 𝑦 ∈ V
24 vex 3193 . . . . . . . . . 10 𝑧 ∈ V
2523, 24brcnv 5275 . . . . . . . . 9 (𝑦𝐹𝑧𝑧𝐹𝑦)
2625eubii 2491 . . . . . . . 8 (∃!𝑧 𝑦𝐹𝑧 ↔ ∃!𝑧 𝑧𝐹𝑦)
27 df-reu 2915 . . . . . . . 8 (∃!𝑥𝐴 𝑦 = 𝐶 ↔ ∃!𝑥(𝑥𝐴𝑦 = 𝐶))
2822, 26, 273bitr4i 292 . . . . . . 7 (∃!𝑧 𝑦𝐹𝑧 ↔ ∃!𝑥𝐴 𝑦 = 𝐶)
2928ralbii 2976 . . . . . 6 (∀𝑦𝐵 ∃!𝑧 𝑦𝐹𝑧 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶)
305, 29bitri 264 . . . . 5 ((𝐹𝐵) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶)
31 relcnv 5472 . . . . . . 7 Rel 𝐹
32 df-rn 5095 . . . . . . . 8 ran 𝐹 = dom 𝐹
33 frn 6020 . . . . . . . 8 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
3432, 33syl5eqssr 3635 . . . . . . 7 (𝐹:𝐴𝐵 → dom 𝐹𝐵)
35 relssres 5406 . . . . . . 7 ((Rel 𝐹 ∧ dom 𝐹𝐵) → (𝐹𝐵) = 𝐹)
3631, 34, 35sylancr 694 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐹)
3736fneq1d 5949 . . . . 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 6104 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
4241pm4.71ri 664 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴𝐵𝐹:𝐴1-1-onto𝐵))
437fmpt 6347 . . 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 2908  ∃!wreu 2910  wss 3560  cop 4161   class class class wbr 4623  {copab 4682  cmpt 4683  ccnv 5083  dom cdm 5084  ran crn 5085  cres 5086  Rel wrel 5089   Fn wfn 5852  wf 5853  1-1-ontowf1o 5856
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 4751  ax-nul 4759  ax-pr 4877
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 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865
This theorem is referenced by:  oaf1o  7603  xpf1o  8082  icoshftf1o  12253  fprodser  14623  dfod2  17921  gsummptf1o  18302  nbusgrf1o0  26192  cusgrfilem2  26273  numclwlk2lem2f1o  27127  f1mptrn  29319  xrmulc1cn  29800  poimirlem4  33084  poimirlem16  33096  poimirlem17  33097  poimirlem19  33099  poimirlem20  33100
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