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Theorem f1oresrab 5673
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 5455 . . . 4 (𝐹:𝐴1-1-onto𝐵 → Fun 𝐹)
3 funcnvcnv 5267 . . . 4 (Fun 𝐹 → Fun 𝐹)
41, 2, 33syl 17 . . 3 (𝜑 → Fun 𝐹)
5 f1ocnv 5466 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐵1-1-onto𝐴)
61, 5syl 14 . . . . . 6 (𝜑𝐹:𝐵1-1-onto𝐴)
7 f1of1 5452 . . . . . 6 (𝐹:𝐵1-1-onto𝐴𝐹:𝐵1-1𝐴)
86, 7syl 14 . . . . 5 (𝜑𝐹:𝐵1-1𝐴)
9 ssrab2 3238 . . . . 5 {𝑦𝐵𝜒} ⊆ 𝐵
10 f1ores 5468 . . . . 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 5114 . . . . . 6 (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}}
14 f1oresrab.3 . . . . . . . . . 10 ((𝜑𝑥𝐴𝑦 = 𝐶) → (𝜒𝜓))
15143expia 1205 . . . . . . . . 9 ((𝜑𝑥𝐴) → (𝑦 = 𝐶 → (𝜒𝜓)))
1615alrimiv 1872 . . . . . . . 8 ((𝜑𝑥𝐴) → ∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)))
17 f1of 5453 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
181, 17syl 14 . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
1912fmpt 5658 . . . . . . . . . 10 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
2018, 19sylibr 134 . . . . . . . . 9 (𝜑 → ∀𝑥𝐴 𝐶𝐵)
2120r19.21bi 2563 . . . . . . . 8 ((𝜑𝑥𝐴) → 𝐶𝐵)
22 elrab3t 2890 . . . . . . . 8 ((∀𝑦(𝑦 = 𝐶 → (𝜒𝜓)) ∧ 𝐶𝐵) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2316, 21, 22syl2anc 411 . . . . . . 7 ((𝜑𝑥𝐴) → (𝐶 ∈ {𝑦𝐵𝜒} ↔ 𝜓))
2423rabbidva 2723 . . . . . 6 (𝜑 → {𝑥𝐴𝐶 ∈ {𝑦𝐵𝜒}} = {𝑥𝐴𝜓})
2513, 24eqtrid 2220 . . . . 5 (𝜑 → (𝐹 “ {𝑦𝐵𝜒}) = {𝑥𝐴𝜓})
26 f1oeq3 5443 . . . . 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 5469 . . 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 5083 . . 3 (𝐹 ↾ {𝑥𝐴𝜓}) = (𝐹 ↾ {𝑥𝐴𝜓})
32 f1oeq1 5441 . . 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 978  wal 1351   = wceq 1353  wcel 2146  wral 2453  {crab 2457  wss 3127  cmpt 4059  ccnv 4619  cres 4622  cima 4623  Fun wfun 5202  wf 5204  1-1wf1 5205  1-1-ontowf1o 5207
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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216
This theorem is referenced by: (None)
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