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Theorem f1ocoima 7321
Description: The composition of two bijections as bijection onto the image of the range of the first bijection. (Contributed by AV, 15-Aug-2025.)
Assertion
Ref Expression
f1ocoima ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → (𝐺𝐹):𝐴1-1-onto→(𝐺𝐵))

Proof of Theorem f1ocoima
StepHypRef Expression
1 f1of1 6845 . . . . . 6 (𝐺:𝐶1-1-onto𝐷𝐺:𝐶1-1𝐷)
21anim1i 615 . . . . 5 ((𝐺:𝐶1-1-onto𝐷𝐵𝐶) → (𝐺:𝐶1-1𝐷𝐵𝐶))
323adant1 1131 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → (𝐺:𝐶1-1𝐷𝐵𝐶))
4 f1ores 6860 . . . 4 ((𝐺:𝐶1-1𝐷𝐵𝐶) → (𝐺𝐵):𝐵1-1-onto→(𝐺𝐵))
53, 4syl 17 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → (𝐺𝐵):𝐵1-1-onto→(𝐺𝐵))
6 simp1 1137 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → 𝐹:𝐴1-1-onto𝐵)
7 f1oco 6869 . . 3 (((𝐺𝐵):𝐵1-1-onto→(𝐺𝐵) ∧ 𝐹:𝐴1-1-onto𝐵) → ((𝐺𝐵) ∘ 𝐹):𝐴1-1-onto→(𝐺𝐵))
85, 6, 7syl2anc 584 . 2 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → ((𝐺𝐵) ∘ 𝐹):𝐴1-1-onto→(𝐺𝐵))
9 f1ofo 6853 . . . . . . 7 (𝐹:𝐴1-1-onto𝐵𝐹:𝐴onto𝐵)
10 forn 6821 . . . . . . 7 (𝐹:𝐴onto𝐵 → ran 𝐹 = 𝐵)
119, 10syl 17 . . . . . 6 (𝐹:𝐴1-1-onto𝐵 → ran 𝐹 = 𝐵)
1211eqimssd 4039 . . . . 5 (𝐹:𝐴1-1-onto𝐵 → ran 𝐹𝐵)
13123ad2ant1 1134 . . . 4 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → ran 𝐹𝐵)
14 cores 6267 . . . . 5 (ran 𝐹𝐵 → ((𝐺𝐵) ∘ 𝐹) = (𝐺𝐹))
1514eqcomd 2742 . . . 4 (ran 𝐹𝐵 → (𝐺𝐹) = ((𝐺𝐵) ∘ 𝐹))
1613, 15syl 17 . . 3 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → (𝐺𝐹) = ((𝐺𝐵) ∘ 𝐹))
1716f1oeq1d 6841 . 2 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → ((𝐺𝐹):𝐴1-1-onto→(𝐺𝐵) ↔ ((𝐺𝐵) ∘ 𝐹):𝐴1-1-onto→(𝐺𝐵)))
188, 17mpbird 257 1 ((𝐹:𝐴1-1-onto𝐵𝐺:𝐶1-1-onto𝐷𝐵𝐶) → (𝐺𝐹):𝐴1-1-onto→(𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wss 3950  ran crn 5684  cres 5685  cima 5686  ccom 5687  1-1wf1 6556  ontowfo 6557  1-1-ontowf1o 6558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566
This theorem is referenced by:  3f1oss1  47060
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