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Theorem f1oresrab 6643
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 6379 . . . 4 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
3 funcnvcnv 6188 . . . 4 (Fun 𝐹 → Fun 𝐹)
41, 2, 33syl 18 . . 3 (𝜑 → Fun 𝐹)
5 f1ocnv 6389 . . . . . 6 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
6 f1of1 6376 . . . . . 6 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵1-1𝐴)
71, 5, 63syl 18 . . . . 5 (𝜑𝐹:𝐵1-1𝐴)
8 ssrab2 3911 . . . . 5 {𝑦𝐵𝜒} ⊆ 𝐵
9 f1ores 6391 . . . . 5 ((𝐹:𝐵1-1𝐴 ∧ {𝑦𝐵𝜒} ⊆ 𝐵) → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}))
107, 8, 9sylancl 582 . . . 4 (𝜑 → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}))
11 f1oresrab.1 . . . . . . 7 𝐹 = (𝑥𝐴𝐶)
1211mptpreima 5868 . . . . . 6 (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}}
13 f1oresrab.3 . . . . . . . . . 10 ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))
14133expia 1156 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑦 = 𝐶 → (𝜒𝜓)))
1514alrimiv 2028 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)))
16 f1of 6377 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
171, 16syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
1811fmpt 6628 . . . . . . . . . 10 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
1917, 18sylibr 226 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
2019r19.21bi 3140 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐶𝐵)
21 elrab3t 3583 . . . . . . . 8 ((∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)) ∧ 𝐶𝐵) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2215, 20, 21syl2anc 581 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2322rabbidva 3400 . . . . . 6 (𝜑 → {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}} = {𝑥𝐴𝜓})
2412, 23syl5eq 2872 . . . . 5 (𝜑 → (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝜓})
25 f1oeq3 6368 . . . . 5 ((𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝜓} → ((𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}) ↔ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}))
2624, 25syl 17 . . . 4 (𝜑 → ((𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}) ↔ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}))
2710, 26mpbid 224 . . 3 (𝜑 → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓})
28 f1orescnv 6392 . . 3 ((Fun 𝐹 ∧ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}) → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
294, 27, 28syl2anc 581 . 2 (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
30 rescnvcnv 5837 . . 3 (𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓})
31 f1oeq1 6366 . . 3 ((𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓}) → ((𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒}))
3230, 31ax-mp 5 . 2 ((𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
3329, 32sylib 210 1 (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1113  wal 1656   = wceq 1658  wcel 2166  wral 3116  {crab 3120  wss 3797  cmpt 4951  ccnv 5340  cres 5343  cima 5344  Fun wfun 6116  wf 6118  1-1wf1 6119  1-1-ontowf1o 6121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pr 5126
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ne 2999  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-ima 5354  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130
This theorem is referenced by:  f1ossf1o  6644  wlknwwlksnbij  27191  wlksnwwlknvbij  27229  wlksnwwlknvbijOLD  27230  clwlknf1oclwwlkn  27450  clwlknf1oclwwlknOLD  27452  clwwlkvbij  27487  clwwlkvbijOLD  27488  rabfodom  29891  fpwrelmapffs  30056  eulerpartlemn  30987  f1oresf1orab  42191
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