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Theorem f1oresf1o2 44308
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 6619 . . . . . . . . . . 11 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴𝐵)
41, 3syl 17 . . . . . . . . . 10 (𝜑𝐹:𝐴𝐵)
54adantr 484 . . . . . . . . 9 ((𝜑𝑥𝐷) → 𝐹:𝐴𝐵)
62sselda 3878 . . . . . . . . 9 ((𝜑𝑥𝐷) → 𝑥𝐴)
75, 6jca 515 . . . . . . . 8 ((𝜑𝑥𝐷) → (𝐹:𝐴𝐵𝑥𝐴))
873adant3 1133 . . . . . . 7 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝐹:𝐴𝐵𝑥𝐴))
9 ffvelrn 6860 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
108, 9syl 17 . . . . . 6 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝐹𝑥) ∈ 𝐵)
11 eleq1 2820 . . . . . . 7 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
12113ad2ant3 1136 . . . . . 6 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
1310, 12mpbid 235 . . . . 5 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → 𝑦𝐵)
14 eqcom 2745 . . . . . . . 8 ((𝐹𝑥) = 𝑦𝑦 = (𝐹𝑥))
15 f1oresf1o2.3 . . . . . . . . . 10 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
1615biimpd 232 . . . . . . . . 9 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷𝜒))
1716ex 416 . . . . . . . 8 (𝜑 → (𝑦 = (𝐹𝑥) → (𝑥𝐷𝜒)))
1814, 17syl5bi 245 . . . . . . 7 (𝜑 → ((𝐹𝑥) = 𝑦 → (𝑥𝐷𝜒)))
1918com23 86 . . . . . 6 (𝜑 → (𝑥𝐷 → ((𝐹𝑥) = 𝑦𝜒)))
20193imp 1112 . . . . 5 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → 𝜒)
2113, 20jca 515 . . . 4 ((𝜑𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → (𝑦𝐵𝜒))
2221rexlimdv3a 3196 . . 3 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 → (𝑦𝐵𝜒)))
23 f1ofo 6626 . . . . . . . 8 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
241, 23syl 17 . . . . . . 7 (𝜑𝐹:𝐴onto𝐵)
25 foelrni 6732 . . . . . . 7 ((𝐹:𝐴onto𝐵𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
2624, 25sylan 583 . . . . . 6 ((𝜑𝑦𝐵) → ∃𝑥𝐴 (𝐹𝑥) = 𝑦)
2726ex 416 . . . . 5 (𝜑 → (𝑦𝐵 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
28 nfv 1920 . . . . . 6 𝑥𝜑
29 nfv 1920 . . . . . . 7 𝑥𝜒
30 nfre1 3216 . . . . . . 7 𝑥𝑥𝐷 (𝐹𝑥) = 𝑦
3129, 30nfim 1902 . . . . . 6 𝑥(𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)
32 rspe 3214 . . . . . . . . . . . . . 14 ((𝑥𝐷 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)
3332expcom 417 . . . . . . . . . . . . 13 ((𝐹𝑥) = 𝑦 → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3433eqcoms 2746 . . . . . . . . . . . 12 (𝑦 = (𝐹𝑥) → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3534adantl 485 . . . . . . . . . . 11 ((𝜑𝑦 = (𝐹𝑥)) → (𝑥𝐷 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3615, 35sylbird 263 . . . . . . . . . 10 ((𝜑𝑦 = (𝐹𝑥)) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
3736ex 416 . . . . . . . . 9 (𝜑 → (𝑦 = (𝐹𝑥) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
3837adantr 484 . . . . . . . 8 ((𝜑𝑥𝐴) → (𝑦 = (𝐹𝑥) → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
3914, 38syl5bi 245 . . . . . . 7 ((𝜑𝑥𝐴) → ((𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
4039ex 416 . . . . . 6 (𝜑 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))))
4128, 31, 40rexlimd 3227 . . . . 5 (𝜑 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
4227, 41syld 47 . . . 4 (𝜑 → (𝑦𝐵 → (𝜒 → ∃𝑥𝐷 (𝐹𝑥) = 𝑦)))
4342impd 414 . . 3 (𝜑 → ((𝑦𝐵𝜒) → ∃𝑥𝐷 (𝐹𝑥) = 𝑦))
4422, 43impbid 215 . 2 (𝜑 → (∃𝑥𝐷 (𝐹𝑥) = 𝑦 ↔ (𝑦𝐵𝜒)))
451, 2, 44f1oresf1o 44307 1 (𝜑 → (𝐹𝐷):𝐷1-1-onto→{𝑦𝐵𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wrex 3054  {crab 3057  wss 3844  cres 5528  wf 6336  ontowfo 6338  1-1-ontowf1o 6339  cfv 6340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5168  ax-nul 5175  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3683  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348
This theorem is referenced by: (None)
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