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Theorem f1ompt 5261
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 = (x A𝐶)
Assertion
Ref Expression
f1ompt (𝐹:A1-1-ontoB ↔ (x A 𝐶 B y B ∃!x A 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 4987 . . . . 5 (𝐹:AB𝐹 Fn A)
2 dff1o4 5075 . . . . . 6 (𝐹:A1-1-ontoB ↔ (𝐹 Fn A 𝐹 Fn B))
32baib 827 . . . . 5 (𝐹 Fn A → (𝐹:A1-1-ontoB𝐹 Fn B))
41, 3syl 14 . . . 4 (𝐹:AB → (𝐹:A1-1-ontoB𝐹 Fn B))
5 fnres 4956 . . . . . 6 ((𝐹B) Fn By B ∃!z y𝐹z)
6 nfcv 2175 . . . . . . . . . 10 xz
7 fmpt.1 . . . . . . . . . . 11 𝐹 = (x A𝐶)
8 nfmpt1 3840 . . . . . . . . . . 11 x(x A𝐶)
97, 8nfcxfr 2172 . . . . . . . . . 10 x𝐹
10 nfcv 2175 . . . . . . . . . 10 xy
116, 9, 10nfbr 3798 . . . . . . . . 9 x z𝐹y
12 nfv 1418 . . . . . . . . 9 z(x A y = 𝐶)
13 breq1 3757 . . . . . . . . . 10 (z = x → (z𝐹yx𝐹y))
14 df-mpt 3810 . . . . . . . . . . . . 13 (x A𝐶) = {⟨x, y⟩ ∣ (x A y = 𝐶)}
157, 14eqtri 2057 . . . . . . . . . . . 12 𝐹 = {⟨x, y⟩ ∣ (x A y = 𝐶)}
1615breqi 3760 . . . . . . . . . . 11 (x𝐹yx{⟨x, y⟩ ∣ (x A y = 𝐶)}y)
17 df-br 3755 . . . . . . . . . . . 12 (x{⟨x, y⟩ ∣ (x A y = 𝐶)}y ↔ ⟨x, y {⟨x, y⟩ ∣ (x A y = 𝐶)})
18 opabid 3984 . . . . . . . . . . . 12 (⟨x, y {⟨x, y⟩ ∣ (x A y = 𝐶)} ↔ (x A y = 𝐶))
1917, 18bitri 173 . . . . . . . . . . 11 (x{⟨x, y⟩ ∣ (x A y = 𝐶)}y ↔ (x A y = 𝐶))
2016, 19bitri 173 . . . . . . . . . 10 (x𝐹y ↔ (x A y = 𝐶))
2113, 20syl6bb 185 . . . . . . . . 9 (z = x → (z𝐹y ↔ (x A y = 𝐶)))
2211, 12, 21cbveu 1921 . . . . . . . 8 (∃!z z𝐹y∃!x(x A y = 𝐶))
23 vex 2554 . . . . . . . . . 10 y V
24 vex 2554 . . . . . . . . . 10 z V
2523, 24brcnv 4460 . . . . . . . . 9 (y𝐹zz𝐹y)
2625eubii 1906 . . . . . . . 8 (∃!z y𝐹z∃!z z𝐹y)
27 df-reu 2307 . . . . . . . 8 (∃!x A y = 𝐶∃!x(x A y = 𝐶))
2822, 26, 273bitr4i 201 . . . . . . 7 (∃!z y𝐹z∃!x A y = 𝐶)
2928ralbii 2324 . . . . . 6 (y B ∃!z y𝐹zy B ∃!x A y = 𝐶)
305, 29bitri 173 . . . . 5 ((𝐹B) Fn By B ∃!x A y = 𝐶)
31 relcnv 4645 . . . . . . 7 Rel 𝐹
32 df-rn 4298 . . . . . . . 8 ran 𝐹 = dom 𝐹
33 frn 4993 . . . . . . . 8 (𝐹:AB → ran 𝐹B)
3432, 33syl5eqssr 2984 . . . . . . 7 (𝐹:AB → dom 𝐹B)
35 relssres 4590 . . . . . . 7 ((Rel 𝐹 dom 𝐹B) → (𝐹B) = 𝐹)
3631, 34, 35sylancr 393 . . . . . 6 (𝐹:AB → (𝐹B) = 𝐹)
3736fneq1d 4930 . . . . 5 (𝐹:AB → ((𝐹B) Fn B𝐹 Fn B))
3830, 37syl5bbr 183 . . . 4 (𝐹:AB → (y B ∃!x A y = 𝐶𝐹 Fn B))
394, 38bitr4d 180 . . 3 (𝐹:AB → (𝐹:A1-1-ontoBy B ∃!x A y = 𝐶))
4039pm5.32i 427 . 2 ((𝐹:AB 𝐹:A1-1-ontoB) ↔ (𝐹:AB y B ∃!x A y = 𝐶))
41 f1of 5067 . . 3 (𝐹:A1-1-ontoB𝐹:AB)
4241pm4.71ri 372 . 2 (𝐹:A1-1-ontoB ↔ (𝐹:AB 𝐹:A1-1-ontoB))
437fmpt 5260 . . 3 (x A 𝐶 B𝐹:AB)
4443anbi1i 431 . 2 ((x A 𝐶 B y B ∃!x A y = 𝐶) ↔ (𝐹:AB y B ∃!x A y = 𝐶))
4540, 42, 443bitr4i 201 1 (𝐹:A1-1-ontoB ↔ (x A 𝐶 B y B ∃!x A y = 𝐶))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  ∃!weu 1897  wral 2300  ∃!wreu 2302  wss 2911  cop 3369   class class class wbr 3754  {copab 3807  cmpt 3808  ccnv 4286  dom cdm 4287  ran crn 4288  cres 4289  Rel wrel 4292   Fn wfn 4839  wf 4840  1-1-ontowf1o 4843
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852
This theorem is referenced by:  icoshftf1o  8577
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