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Theorem f1oresrab 5272
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 = (x A𝐶)
f1oresrab.2 (φ𝐹:A1-1-ontoB)
f1oresrab.3 ((φ x A y = 𝐶) → (χψ))
Assertion
Ref Expression
f1oresrab (φ → (𝐹 ↾ {x Aψ}):{x Aψ}–1-1-onto→{y Bχ})
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 5071 . . . 4 (𝐹:A1-1-ontoB → Fun 𝐹)
3 funcnvcnv 4901 . . . 4 (Fun 𝐹 → Fun 𝐹)
41, 2, 33syl 17 . . 3 (φ → Fun 𝐹)
5 f1ocnv 5082 . . . . . . 7 (𝐹:A1-1-ontoB𝐹:B1-1-ontoA)
61, 5syl 14 . . . . . 6 (φ𝐹:B1-1-ontoA)
7 f1of1 5068 . . . . . 6 (𝐹:B1-1-ontoA𝐹:B1-1A)
86, 7syl 14 . . . . 5 (φ𝐹:B1-1A)
9 ssrab2 3019 . . . . 5 {y Bχ} ⊆ B
10 f1ores 5084 . . . . 5 ((𝐹:B1-1A {y Bχ} ⊆ B) → (𝐹 ↾ {y Bχ}):{y Bχ}–1-1-onto→(𝐹 “ {y Bχ}))
118, 9, 10sylancl 392 . . . 4 (φ → (𝐹 ↾ {y Bχ}):{y Bχ}–1-1-onto→(𝐹 “ {y Bχ}))
12 f1oresrab.1 . . . . . . 7 𝐹 = (x A𝐶)
1312mptpreima 4757 . . . . . 6 (𝐹 “ {y Bχ}) = {x A𝐶 {y Bχ}}
14 f1oresrab.3 . . . . . . . . . 10 ((φ x A y = 𝐶) → (χψ))
15143expia 1105 . . . . . . . . 9 ((φ x A) → (y = 𝐶 → (χψ)))
1615alrimiv 1751 . . . . . . . 8 ((φ x A) → y(y = 𝐶 → (χψ)))
17 f1of 5069 . . . . . . . . . . 11 (𝐹:A1-1-ontoB𝐹:AB)
181, 17syl 14 . . . . . . . . . 10 (φ𝐹:AB)
1912fmpt 5262 . . . . . . . . . 10 (x A 𝐶 B𝐹:AB)
2018, 19sylibr 137 . . . . . . . . 9 (φx A 𝐶 B)
2120r19.21bi 2401 . . . . . . . 8 ((φ x A) → 𝐶 B)
22 elrab3t 2691 . . . . . . . 8 ((y(y = 𝐶 → (χψ)) 𝐶 B) → (𝐶 {y Bχ} ↔ ψ))
2316, 21, 22syl2anc 391 . . . . . . 7 ((φ x A) → (𝐶 {y Bχ} ↔ ψ))
2423rabbidva 2542 . . . . . 6 (φ → {x A𝐶 {y Bχ}} = {x Aψ})
2513, 24syl5eq 2081 . . . . 5 (φ → (𝐹 “ {y Bχ}) = {x Aψ})
26 f1oeq3 5062 . . . . 5 ((𝐹 “ {y Bχ}) = {x Aψ} → ((𝐹 ↾ {y Bχ}):{y Bχ}–1-1-onto→(𝐹 “ {y Bχ}) ↔ (𝐹 ↾ {y Bχ}):{y Bχ}–1-1-onto→{x Aψ}))
2725, 26syl 14 . . . 4 (φ → ((𝐹 ↾ {y Bχ}):{y Bχ}–1-1-onto→(𝐹 “ {y Bχ}) ↔ (𝐹 ↾ {y Bχ}):{y Bχ}–1-1-onto→{x Aψ}))
2811, 27mpbid 135 . . 3 (φ → (𝐹 ↾ {y Bχ}):{y Bχ}–1-1-onto→{x Aψ})
29 f1orescnv 5085 . . 3 ((Fun 𝐹 (𝐹 ↾ {y Bχ}):{y Bχ}–1-1-onto→{x Aψ}) → (𝐹 ↾ {x Aψ}):{x Aψ}–1-1-onto→{y Bχ})
304, 28, 29syl2anc 391 . 2 (φ → (𝐹 ↾ {x Aψ}):{x Aψ}–1-1-onto→{y Bχ})
31 rescnvcnv 4726 . . 3 (𝐹 ↾ {x Aψ}) = (𝐹 ↾ {x Aψ})
32 f1oeq1 5060 . . 3 ((𝐹 ↾ {x Aψ}) = (𝐹 ↾ {x Aψ}) → ((𝐹 ↾ {x Aψ}):{x Aψ}–1-1-onto→{y Bχ} ↔ (𝐹 ↾ {x Aψ}):{x Aψ}–1-1-onto→{y Bχ}))
3331, 32ax-mp 7 . 2 ((𝐹 ↾ {x Aψ}):{x Aψ}–1-1-onto→{y Bχ} ↔ (𝐹 ↾ {x Aψ}):{x Aψ}–1-1-onto→{y Bχ})
3430, 33sylib 127 1 (φ → (𝐹 ↾ {x Aψ}):{x Aψ}–1-1-onto→{y Bχ})
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884  wal 1240   = wceq 1242   wcel 1390  wral 2300  {crab 2304  wss 2911  cmpt 3809  ccnv 4287  cres 4290  cima 4291  Fun wfun 4839  wf 4841  1-1wf1 4842  1-1-ontowf1o 4844
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853
This theorem is referenced by: (None)
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