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Theorem f1co 6627
Description: Composition of one-to-one functions when the codomain of the first matches the domain of the second. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by NM, 28-May-1998.) (Proof shortened by AV, 20-Sep-2024.)
Assertion
Ref Expression
f1co ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)

Proof of Theorem f1co
StepHypRef Expression
1 f1cof1 6626 . 2 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):(𝐺𝐵)–1-1𝐶)
2 f1f 6615 . . . . . 6 (𝐺:𝐴1-1𝐵𝐺:𝐴𝐵)
3 fimacnv 6567 . . . . . 6 (𝐺:𝐴𝐵 → (𝐺𝐵) = 𝐴)
42, 3syl 17 . . . . 5 (𝐺:𝐴1-1𝐵 → (𝐺𝐵) = 𝐴)
54adantl 485 . . . 4 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐺𝐵) = 𝐴)
65eqcomd 2743 . . 3 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → 𝐴 = (𝐺𝐵))
7 f1eq2 6611 . . 3 (𝐴 = (𝐺𝐵) → ((𝐹𝐺):𝐴1-1𝐶 ↔ (𝐹𝐺):(𝐺𝐵)–1-1𝐶))
86, 7syl 17 . 2 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → ((𝐹𝐺):𝐴1-1𝐶 ↔ (𝐹𝐺):(𝐺𝐵)–1-1𝐶))
91, 8mpbird 260 1 ((𝐹:𝐵1-1𝐶𝐺:𝐴1-1𝐵) → (𝐹𝐺):𝐴1-1𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  ccnv 5550  cima 5554  ccom 5555  wf 6376  1-1wf1 6377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385
This theorem is referenced by:  f1oco  6683  f1cofveqaeqALT  7071  tposf12  7993  domtr  8681  dfac12lem2  9758  fin23lem28  9954  pwfseqlem5  10277  cofth  17442  injsubmefmnd  18324  gsumzf1o  19297  cycpmconjv  31128  erdsze2lem2  32879  fcoresf1b  44236  fundcmpsurinjpreimafv  44533
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