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Theorem f1oresf1o2 45643
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 6789 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
41, 3syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
54adantr 481 . . . . . . . . 9 ((𝜑𝑥𝐷) → 𝐹:𝐴𝐵)
62sselda 3947 . . . . . . . . 9 ((𝜑𝑥𝐷) → 𝑥𝐴)
75, 6jca 512 . . . . . . . 8 ((𝜑𝑥𝐷) → (𝐹:𝐴𝐵𝑥𝐴))
873adant3 1132 . . . . . . 7 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝐹:𝐴𝐵𝑥𝐴))
9 ffvelcdm 7037 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
108, 9syl 17 . . . . . 6 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
11 eleq1 2820 . . . . . . 7 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
12113ad2ant3 1135 . . . . . 6 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
1310, 12mpbid 231 . . . . 5 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
14 eqcom 2738 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
15 f1oresf1o2.3 . . . . . . . . . 10 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
1615biimpd 228 . . . . . . . . 9 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
1716ex 413 . . . . . . . 8 (𝜑 → (𝑦 = (𝐹𝑥) → (𝑥𝐷𝜒)))
1814, 17biimtrid 241 . . . . . . 7 (𝜑 → ((𝐹𝑥) = 𝑦 → (𝑥𝐷𝜒)))
1918com23 86 . . . . . 6 (𝜑 → (𝑥𝐷 → ((𝐹𝑥) = 𝑦𝜒)))
20193imp 1111 . . . . 5 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → 𝜒)
2113, 20jca 512 . . . 4 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝑦𝐵𝜒))
2221rexlimdv3a 3152 . . 3 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 → (𝑦𝐵𝜒)))
23 f1ofo 6796 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
241, 23syl 17 . . . . . . 7 (𝜑𝐹:𝐴onto𝐵)
25 foelcdmi 6909 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
2624, 25sylan 580 . . . . . 6 ((𝜑𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
2726ex 413 . . . . 5 (𝜑 → (𝑦𝐵 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
28 nfv 1917 . . . . . 6 𝑥𝜑
29 nfv 1917 . . . . . . 7 𝑥𝜒
30 nfre1 3266 . . . . . . 7 𝑥𝑥𝐷 (𝐹𝑥) = 𝑦
3129, 30nfim 1899 . . . . . 6 𝑥(𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)
32 rspe 3230 . . . . . . . . . . . . . 14 ((𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)
3332expcom 414 . . . . . . . . . . . . 13 ((𝐹𝑥) = 𝑦 → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3433eqcoms 2739 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3534adantl 482 . . . . . . . . . . 11 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3615, 35sylbird 259 . . . . . . . . . 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, 38biimtrid 241 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
4039ex 413 . . . . . 6 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))))
4128, 31, 40rexlimd 3247 . . . . 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 211 . 2 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))
451, 2, 44f1oresf1o 45642 1 (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3069  {crab 3405  wss 3913  cres 5640  wf 6497  ontowfo 6499  1-1-ontowf1o 6500  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509
This theorem is referenced by: (None)
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