Theorem f1co 2783

Description: Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
f1co	⊢ ((F:B–1-1→C ∧ G:A–1-1→B) → (F ∘ G):A–1-1→C)

Proof of Theorem f1co

Step	Hyp	Ref	Expression
1		fco 2760	. . . 4 ⊢ ((F:B–→C ∧ G:A–→B) → (F ∘ G):A–→C)
2		funco 2696	. . . . . 6 ⊢ ((Fun ◡G ∧ Fun ◡F) → Fun (◡G ∘ ◡F))
3		cnvco 2520	. . . . . . 7 ⊢ ◡(F ∘ G) = (◡G ∘ ◡F)
4		funeq 2683	. . . . . . 7 ⊢ (◡(F ∘ G) = (◡G ∘ ◡F) → (Fun ◡(F ∘ G) ↔ Fun (◡G ∘ ◡F)))
5	3, 4	ax-mp 6	. . . . . 6 ⊢ (Fun ◡(F ∘ G) ↔ Fun (◡G ∘ ◡F))
6	2, 5	sylibr 175	. . . . 5 ⊢ ((Fun ◡G ∧ Fun ◡F) → Fun ◡(F ∘ G))
7	6	ancoms 334	. . . 4 ⊢ ((Fun ◡F ∧ Fun ◡G) → Fun ◡(F ∘ G))
8	1, 7	anim12i 268	. . 3 ⊢ (((F:B–→C ∧ G:A–→B) ∧ (Fun ◡F ∧ Fun ◡G)) → ((F ∘ G):A–→C ∧ Fun ◡(F ∘ G)))
9	8	an4s 390	. 2 ⊢ (((F:B–→C ∧ Fun ◡F) ∧ (G:A–→B ∧ Fun ◡G)) → ((F ∘ G):A–→C ∧ Fun ◡(F ∘ G)))
10		df-f1 2435	. . 3 ⊢ (F:B–1-1→C ↔ (F:B–→C ∧ Fun ◡F))
11		df-f1 2435	. . 3 ⊢ (G:A–1-1→B ↔ (G:A–→B ∧ Fun ◡G))
12	10, 11	anbi12i 369	. 2 ⊢ ((F:B–1-1→C ∧ G:A–1-1→B) ↔ ((F:B–→C ∧ Fun ◡F) ∧ (G:A–→B ∧ Fun ◡G)))
13		df-f1 2435	. 2 ⊢ ((F ∘ G):A–1-1→C ↔ ((F ∘ G):A–→C ∧ Fun ◡(F ∘ G)))
14	9, 12, 13	3imtr4 192	1 ⊢ ((F:B–1-1→C ∧ G:A–1-1→B) → (F ∘ G):A–1-1→C)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091  ^◡ccnv 2409   ∘ ccom 2414  Fun wfun 2416  –→wf 2418  –1-1→wf1 2419

This theorem is referenced by:  domtr 3320

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435

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