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Mirrors > Home > NFE Home > Th. List > f1co | GIF version |
Description: Composition of one-to-one functions. Exercise 30 of [TakeutiZaring] p. 25. (Contributed by set.mm contributors, 28-May-1998.) |
Ref | Expression |
---|---|
f1co | ⊢ ((F:B–1-1→C ∧ G:A–1-1→B) → (F ∘ G):A–1-1→C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fco 5231 | . . . 4 ⊢ ((F:B–→C ∧ G:A–→B) → (F ∘ G):A–→C) | |
2 | funco 5142 | . . . . . 6 ⊢ ((Fun ◡G ∧ Fun ◡F) → Fun (◡G ∘ ◡F)) | |
3 | cnvco 4894 | . . . . . . 7 ⊢ ◡(F ∘ G) = (◡G ∘ ◡F) | |
4 | 3 | funeqi 5128 | . . . . . 6 ⊢ (Fun ◡(F ∘ G) ↔ Fun (◡G ∘ ◡F)) |
5 | 2, 4 | sylibr 203 | . . . . 5 ⊢ ((Fun ◡G ∧ Fun ◡F) → Fun ◡(F ∘ G)) |
6 | 5 | ancoms 439 | . . . 4 ⊢ ((Fun ◡F ∧ Fun ◡G) → Fun ◡(F ∘ G)) |
7 | 1, 6 | anim12i 549 | . . 3 ⊢ (((F:B–→C ∧ G:A–→B) ∧ (Fun ◡F ∧ Fun ◡G)) → ((F ∘ G):A–→C ∧ Fun ◡(F ∘ G))) |
8 | 7 | an4s 799 | . 2 ⊢ (((F:B–→C ∧ Fun ◡F) ∧ (G:A–→B ∧ Fun ◡G)) → ((F ∘ G):A–→C ∧ Fun ◡(F ∘ G))) |
9 | df-f1 4792 | . . 3 ⊢ (F:B–1-1→C ↔ (F:B–→C ∧ Fun ◡F)) | |
10 | df-f1 4792 | . . 3 ⊢ (G:A–1-1→B ↔ (G:A–→B ∧ Fun ◡G)) | |
11 | 9, 10 | anbi12i 678 | . 2 ⊢ ((F:B–1-1→C ∧ G:A–1-1→B) ↔ ((F:B–→C ∧ Fun ◡F) ∧ (G:A–→B ∧ Fun ◡G))) |
12 | df-f1 4792 | . 2 ⊢ ((F ∘ G):A–1-1→C ↔ ((F ∘ G):A–→C ∧ Fun ◡(F ∘ G))) | |
13 | 8, 11, 12 | 3imtr4i 257 | 1 ⊢ ((F:B–1-1→C ∧ G:A–1-1→B) → (F ∘ G):A–1-1→C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∘ ccom 4721 ◡ccnv 4771 Fun wfun 4775 –→wf 4777 –1-1→wf1 4778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 |
This theorem is referenced by: f1oco 5308 nclenc 6222 |
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