MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1oresrab Structured version   Visualization version   GIF version

Theorem f1oresrab 7130
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 6835 . . . 4 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
3 funcnvcnv 6614 . . . 4 (Fun 𝐹 → Fun 𝐹)
41, 2, 33syl 18 . . 3 (𝜑 → Fun 𝐹)
5 f1ocnv 6845 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1of1 6832 . . . . . 6 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵1-1𝐴)
71, 5, 63syl 18 . . . . 5 (𝜑𝐹:𝐵1-1𝐴)
8 ssrab2 4073 . . . . 5 {𝑦𝐵𝜒} ⊆ 𝐵
9 f1ores 6847 . . . . 5 ((𝐹:𝐵1-1𝐴 ∧ {𝑦𝐵𝜒} ⊆ 𝐵) → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}))
107, 8, 9sylancl 585 . . . 4 (𝜑 → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}))
11 f1oresrab.1 . . . . . . 7 𝐹 = (𝑥𝐴𝐶)
1211mptpreima 6236 . . . . . 6 (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}}
13 f1oresrab.3 . . . . . . . . . 10 ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))
14133expia 1119 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑦 = 𝐶 → (𝜒𝜓)))
1514alrimiv 1923 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)))
16 f1of 6833 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
171, 16syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
1811fmpt 7114 . . . . . . . . . 10 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
1917, 18sylibr 233 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
2019r19.21bi 3244 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐶𝐵)
21 elrab3t 3680 . . . . . . . 8 ((∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)) ∧ 𝐶𝐵) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2215, 20, 21syl2anc 583 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2322rabbidva 3435 . . . . . 6 (𝜑 → {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}} = {𝑥𝐴𝜓})
2412, 23eqtrid 2780 . . . . 5 (𝜑 → (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝜓})
2524f1oeq3d 6830 . . . 4 (𝜑 → ((𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}) ↔ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}))
2610, 25mpbid 231 . . 3 (𝜑 → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓})
27 f1orescnv 6848 . . 3 ((Fun 𝐹 ∧ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}) → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
284, 26, 27syl2anc 583 . 2 (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
29 rescnvcnv 6202 . . 3 (𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓})
30 f1oeq1 6821 . . 3 ((𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓}) → ((𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒}))
3129, 30ax-mp 5 . 2 ((𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
3228, 31sylib 217 1 (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1085  wal 1532   = wceq 1534  wcel 2099  wral 3057  {crab 3428  wss 3945  cmpt 5225  ccnv 5671  cres 5674  cima 5675  Fun wfun 6536  wf 6538  1-1wf1 6539  1-1-ontowf1o 6541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549
This theorem is referenced by:  f1ossf1o  7131  wlknwwlksnbij  29692  wlksnwwlknvbij  29712  clwlknf1oclwwlkn  29887  clwwlkvbij  29916  rabfodom  32294  fpwrelmapffs  32510  eulerpartlemn  33995  f1oresf1orab  46663
  Copyright terms: Public domain W3C validator