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Theorem f1ompt 6525
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 6186 . . . . 5 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 dff1o4 6287 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹 Fn 𝐴𝐹 Fn 𝐵))
32baib 519 . . . . 5 (𝐹 Fn 𝐴 → (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐵))
41, 3syl 17 . . . 4 (𝐹:𝐴𝐵 → (𝐹:𝐴1-1-onto𝐵𝐹 Fn 𝐵))
5 fnres 6148 . . . . . 6 ((𝐹𝐵) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑧 𝑦𝐹𝑧)
6 nfcv 2913 . . . . . . . . . 10 𝑥𝑧
7 fmpt.1 . . . . . . . . . . 11 𝐹 = (𝑥𝐴𝐶)
8 nfmpt1 4882 . . . . . . . . . . 11 𝑥(𝑥𝐴𝐶)
97, 8nfcxfr 2911 . . . . . . . . . 10 𝑥𝐹
10 nfcv 2913 . . . . . . . . . 10 𝑥𝑦
116, 9, 10nfbr 4834 . . . . . . . . 9 𝑥 𝑧𝐹𝑦
12 nfv 1995 . . . . . . . . 9 𝑧(𝑥𝐴𝑦 = 𝐶)
13 breq1 4790 . . . . . . . . . 10 (𝑧 = 𝑥 → (𝑧𝐹𝑦𝑥𝐹𝑦))
14 df-mpt 4865 . . . . . . . . . . . . 13 (𝑥𝐴𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
157, 14eqtri 2793 . . . . . . . . . . . 12 𝐹 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}
1615breqi 4793 . . . . . . . . . . 11 (𝑥𝐹𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦)
17 df-br 4788 . . . . . . . . . . . 12 (𝑥{⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐶)})
18 opabid 5116 . . . . . . . . . . . 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 2654 . . . . . . . 8 (∃!𝑧 𝑧𝐹𝑦 ↔ ∃!𝑥(𝑥𝐴𝑦 = 𝐶))
23 vex 3354 . . . . . . . . . 10 𝑦 ∈ V
24 vex 3354 . . . . . . . . . 10 𝑧 ∈ V
2523, 24brcnv 5444 . . . . . . . . 9 (𝑦𝐹𝑧𝑧𝐹𝑦)
2625eubii 2640 . . . . . . . 8 (∃!𝑧 𝑦𝐹𝑧 ↔ ∃!𝑧 𝑧𝐹𝑦)
27 df-reu 3068 . . . . . . . 8 (∃!𝑥𝐴 𝑦 = 𝐶 ↔ ∃!𝑥(𝑥𝐴𝑦 = 𝐶))
2822, 26, 273bitr4i 292 . . . . . . 7 (∃!𝑧 𝑦𝐹𝑧 ↔ ∃!𝑥𝐴 𝑦 = 𝐶)
2928ralbii 3129 . . . . . 6 (∀𝑦𝐵 ∃!𝑧 𝑦𝐹𝑧 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶)
305, 29bitri 264 . . . . 5 ((𝐹𝐵) Fn 𝐵 ↔ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶)
31 relcnv 5645 . . . . . . 7 Rel 𝐹
32 df-rn 5261 . . . . . . . 8 ran 𝐹 = dom 𝐹
33 frn 6194 . . . . . . . 8 (𝐹:𝐴𝐵 → ran 𝐹𝐵)
3432, 33syl5eqssr 3800 . . . . . . 7 (𝐹:𝐴𝐵 → dom 𝐹𝐵)
35 relssres 5579 . . . . . . 7 ((Rel 𝐹 ∧ dom 𝐹𝐵) → (𝐹𝐵) = 𝐹)
3631, 34, 35sylancr 569 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐹)
3736fneq1d 6122 . . . . 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 558 . 2 ((𝐹:𝐴𝐵𝐹:𝐴1-1-onto𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑦𝐵 ∃!𝑥𝐴 𝑦 = 𝐶))
41 f1of 6279 . . 3 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
4241pm4.71ri 544 . 2 (𝐹:𝐴1-1-onto𝐵 ↔ (𝐹:𝐴𝐵𝐹:𝐴1-1-onto𝐵))
437fmpt 6524 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
4443anbi1i 604 . 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 382   = wceq 1631  wcel 2145  ∃!weu 2618  wral 3061  ∃!wreu 3063  wss 3724  cop 4323   class class class wbr 4787  {copab 4847  cmpt 4864  ccnv 5249  dom cdm 5250  ran crn 5251  cres 5252  Rel wrel 5255   Fn wfn 6027  wf 6028  1-1-ontowf1o 6031
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-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  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 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  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 5995  df-fun 6034  df-fn 6035  df-f 6036  df-f1 6037  df-fo 6038  df-f1o 6039  df-fv 6040
This theorem is referenced by:  oaf1o  7798  xpf1o  8279  icoshftf1o  12503  fprodser  14887  dfod2  18189  gsummptf1o  18570  nbusgrf1o0  26495  cusgrfilem2  26588  numclwlk2lem2f1o  27571  numclwlk2lem2f1oOLD  27578  f1mptrn  29776  xrmulc1cn  30317  poimirlem4  33747  poimirlem16  33759  poimirlem17  33760  poimirlem19  33762  poimirlem20  33763
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