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Theorem f1oresf1o2 43353
Description: Build a bijection by restricting the domain of a bijection. (Contributed by AV, 31-Jul-2022.)
Hypotheses
Ref Expression
f1oresf1o2.1 (𝜑𝐹:𝐴1-1-onto𝐵)
f1oresf1o2.2 (𝜑𝐷𝐴)
f1oresf1o2.3 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
Assertion
Ref Expression
f1oresf1o2 (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑦,𝐷   𝑥,𝐹,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥
Allowed substitution hints:   𝜒(𝑦)   𝐴(𝑦)   𝐵(𝑦)

Proof of Theorem f1oresf1o2
StepHypRef Expression
1 f1oresf1o2.1 . 2 (𝜑𝐹:𝐴1-1-onto𝐵)
2 f1oresf1o2.2 . 2 (𝜑𝐷𝐴)
3 f1of 6611 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
41, 3syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
54adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐷) → 𝐹:𝐴𝐵)
62sselda 3970 . . . . . . . . 9 ((𝜑𝑥𝐷) → 𝑥𝐴)
75, 6jca 512 . . . . . . . 8 ((𝜑𝑥𝐷) → (𝐹:𝐴𝐵𝑥𝐴))
873adant3 1126 . . . . . . 7 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝐹:𝐴𝐵𝑥𝐴))
9 ffvelrn 6844 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
108, 9syl 17 . . . . . 6 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
11 eleq1 2904 . . . . . . 7 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
12113ad2ant3 1129 . . . . . 6 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
1310, 12mpbid 233 . . . . 5 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
14 eqcom 2831 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
15 f1oresf1o2.3 . . . . . . . . . 10 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
1615biimpd 230 . . . . . . . . 9 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
1716ex 413 . . . . . . . 8 (𝜑 → (𝑦 = (𝐹𝑥) → (𝑥𝐷𝜒)))
1814, 17syl5bi 243 . . . . . . 7 (𝜑 → ((𝐹𝑥) = 𝑦 → (𝑥𝐷𝜒)))
1918com23 86 . . . . . 6 (𝜑 → (𝑥𝐷 → ((𝐹𝑥) = 𝑦𝜒)))
20193imp 1105 . . . . 5 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → 𝜒)
2113, 20jca 512 . . . 4 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝑦𝐵𝜒))
2221rexlimdv3a 3290 . . 3 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 → (𝑦𝐵𝜒)))
23 f1ofo 6618 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
241, 23syl 17 . . . . . . 7 (𝜑𝐹:𝐴onto𝐵)
25 foelrni 6723 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
2624, 25sylan 580 . . . . . 6 ((𝜑𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
2726ex 413 . . . . 5 (𝜑 → (𝑦𝐵 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
28 nfv 1908 . . . . . 6 𝑥𝜑
29 nfv 1908 . . . . . . 7 𝑥𝜒
30 nfre1 3310 . . . . . . 7 𝑥𝑥𝐷 (𝐹𝑥) = 𝑦
3129, 30nfim 1890 . . . . . 6 𝑥(𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)
32 rspe 3308 . . . . . . . . . . . . . 14 ((𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)
3332expcom 414 . . . . . . . . . . . . 13 ((𝐹𝑥) = 𝑦 → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3433eqcoms 2832 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3534adantl 482 . . . . . . . . . . 11 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3615, 35sylbird 261 . . . . . . . . . 10 ((𝜑𝑦 = (𝐹𝑥)) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3736ex 413 . . . . . . . . 9 (𝜑 → (𝑦 = (𝐹𝑥) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
3837adantr 481 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
3914, 38syl5bi 243 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
4039ex 413 . . . . . 6 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))))
4128, 31, 40rexlimd 3321 . . . . 5 (𝜑 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
4227, 41syld 47 . . . 4 (𝜑 → (𝑦𝐵 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
4342impd 411 . . 3 (𝜑 → ((𝑦𝐵𝜒) → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
4422, 43impbid 213 . 2 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))
451, 2, 44f1oresf1o 43352 1 (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wcel 2106  wrex 3143  {crab 3146  wss 3939  cres 5555  wf 6347  ontowfo 6349  1-1-ontowf1o 6350  cfv 6351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359
This theorem is referenced by: (None)
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