ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  f1oresrab GIF version

Theorem f1oresrab 5539
Description: Build a bijection between restricted abstract builders, given a bijection between the base classes, deduction version. (Contributed by Thierry Arnoux, 17-Aug-2018.)
Hypotheses
Ref Expression
f1oresrab.1 𝐹 = (𝑥𝐴𝐶)
f1oresrab.2 (𝜑𝐹:𝐴1-1-onto𝐵)
f1oresrab.3 ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))
Assertion
Ref Expression
f1oresrab (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝜑,𝑥,𝑦   𝜓,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑦)   𝐶(𝑥)   𝐹(𝑥,𝑦)

Proof of Theorem f1oresrab
StepHypRef Expression
1 f1oresrab.2 . . . 4 (𝜑𝐹:𝐴1-1-onto𝐵)
2 f1ofun 5325 . . . 4 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
3 funcnvcnv 5140 . . . 4 (Fun 𝐹 → Fun 𝐹)
41, 2, 33syl 17 . . 3 (𝜑 → Fun 𝐹)
5 f1ocnv 5336 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
61, 5syl 14 . . . . . 6 (𝜑𝐹:𝐵1-1-onto𝐴)
7 f1of1 5322 . . . . . 6 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵1-1𝐴)
86, 7syl 14 . . . . 5 (𝜑𝐹:𝐵1-1𝐴)
9 ssrab2 3148 . . . . 5 {𝑦𝐵𝜒} ⊆ 𝐵
10 f1ores 5338 . . . . 5 ((𝐹:𝐵1-1𝐴 ∧ {𝑦𝐵𝜒} ⊆ 𝐵) → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}))
118, 9, 10sylancl 407 . . . 4 (𝜑 → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}))
12 f1oresrab.1 . . . . . . 7 𝐹 = (𝑥𝐴𝐶)
1312mptpreima 4990 . . . . . 6 (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}}
14 f1oresrab.3 . . . . . . . . . 10 ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))
15143expia 1166 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑦 = 𝐶 → (𝜒𝜓)))
1615alrimiv 1828 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)))
17 f1of 5323 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
181, 17syl 14 . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
1912fmpt 5524 . . . . . . . . . 10 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
2018, 19sylibr 133 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
2120r19.21bi 2494 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐶𝐵)
22 elrab3t 2808 . . . . . . . 8 ((∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)) ∧ 𝐶𝐵) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2316, 21, 22syl2anc 406 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2423rabbidva 2645 . . . . . 6 (𝜑 → {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}} = {𝑥𝐴𝜓})
2513, 24syl5eq 2159 . . . . 5 (𝜑 → (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝜓})
26 f1oeq3 5316 . . . . 5 ((𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝜓} → ((𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}) ↔ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}))
2725, 26syl 14 . . . 4 (𝜑 → ((𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}) ↔ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}))
2811, 27mpbid 146 . . 3 (𝜑 → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓})
29 f1orescnv 5339 . . 3 ((Fun 𝐹 ∧ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}) → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
304, 28, 29syl2anc 406 . 2 (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
31 rescnvcnv 4959 . . 3 (𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓})
32 f1oeq1 5314 . . 3 ((𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓}) → ((𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒}))
3331, 32ax-mp 7 . 2 ((𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
3430, 33sylib 121 1 (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  w3a 945  wal 1312   = wceq 1314  wcel 1463  wral 2390  {crab 2394  wss 3037  cmpt 3949  ccnv 4498  cres 4501  cima 4502  Fun wfun 5075  wf 5077  1-1wf1 5078  1-1-ontowf1o 5080
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-rab 2399  df-v 2659  df-sbc 2879  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator