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Theorem f1oresrab 5702
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 5482 . . . 4 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
3 funcnvcnv 5294 . . . 4 (Fun 𝐹 → Fun 𝐹)
41, 2, 33syl 17 . . 3 (𝜑 → Fun 𝐹)
5 f1ocnv 5493 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
61, 5syl 14 . . . . . 6 (𝜑𝐹:𝐵1-1-onto𝐴)
7 f1of1 5479 . . . . . 6 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵1-1𝐴)
86, 7syl 14 . . . . 5 (𝜑𝐹:𝐵1-1𝐴)
9 ssrab2 3255 . . . . 5 {𝑦𝐵𝜒} ⊆ 𝐵
10 f1ores 5495 . . . . 5 ((𝐹:𝐵1-1𝐴 ∧ {𝑦𝐵𝜒} ⊆ 𝐵) → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}))
118, 9, 10sylancl 413 . . . 4 (𝜑 → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→(𝐹 “ {𝑦𝐵𝜒}))
12 f1oresrab.1 . . . . . . 7 𝐹 = (𝑥𝐴𝐶)
1312mptpreima 5140 . . . . . 6 (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}}
14 f1oresrab.3 . . . . . . . . . 10 ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))
15143expia 1207 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑦 = 𝐶 → (𝜒𝜓)))
1615alrimiv 1885 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)))
17 f1of 5480 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
181, 17syl 14 . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
1912fmpt 5687 . . . . . . . . . 10 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
2018, 19sylibr 134 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
2120r19.21bi 2578 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐶𝐵)
22 elrab3t 2907 . . . . . . . 8 ((∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)) ∧ 𝐶𝐵) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2316, 21, 22syl2anc 411 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2423rabbidva 2740 . . . . . 6 (𝜑 → {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}} = {𝑥𝐴𝜓})
2513, 24eqtrid 2234 . . . . 5 (𝜑 → (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝜓})
26 f1oeq3 5470 . . . . 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 147 . . 3 (𝜑 → (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓})
29 f1orescnv 5496 . . 3 ((Fun 𝐹 ∧ (𝐹 ↾ {𝑦𝐵𝜒}):{𝑦𝐵𝜒}–1-1-onto→{𝑥𝐴𝜓}) → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
304, 28, 29syl2anc 411 . 2 (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
31 rescnvcnv 5109 . . 3 (𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓})
32 f1oeq1 5468 . . 3 ((𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓}) → ((𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒}))
3331, 32ax-mp 5 . 2 ((𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒} ↔ (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
3430, 33sylib 122 1 (𝜑 → (𝐹 ↾ {𝑥𝐴𝜓}):{𝑥𝐴𝜓}–1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wal 1362   = wceq 1364  wcel 2160  wral 2468  {crab 2472  wss 3144  cmpt 4079  ccnv 4643  cres 4646  cima 4647  Fun wfun 5229  wf 5231  1-1wf1 5232  1-1-ontowf1o 5234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243
This theorem is referenced by: (None)
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