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Mirrors > Home > NFE Home > Th. List > fco | GIF version |
Description: Composition of two mappings. (The proof was shortened by Andrew Salmon, 17-Sep-2011.) (Contributed by set.mm contributors, 29-Aug-1999.) (Revised by set.mm contributors, 18-Sep-2011.) |
Ref | Expression |
---|---|
fco | ⊢ ((F:B–→C ∧ G:A–→B) → (F ∘ G):A–→C) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnco 5191 | . . . . . 6 ⊢ ((F Fn B ∧ G Fn A ∧ ran G ⊆ B) → (F ∘ G) Fn A) | |
2 | 1 | 3expib 1154 | . . . . 5 ⊢ (F Fn B → ((G Fn A ∧ ran G ⊆ B) → (F ∘ G) Fn A)) |
3 | 2 | adantr 451 | . . . 4 ⊢ ((F Fn B ∧ ran F ⊆ C) → ((G Fn A ∧ ran G ⊆ B) → (F ∘ G) Fn A)) |
4 | rncoss 4972 | . . . . . 6 ⊢ ran (F ∘ G) ⊆ ran F | |
5 | sstr 3280 | . . . . . 6 ⊢ ((ran (F ∘ G) ⊆ ran F ∧ ran F ⊆ C) → ran (F ∘ G) ⊆ C) | |
6 | 4, 5 | mpan 651 | . . . . 5 ⊢ (ran F ⊆ C → ran (F ∘ G) ⊆ C) |
7 | 6 | adantl 452 | . . . 4 ⊢ ((F Fn B ∧ ran F ⊆ C) → ran (F ∘ G) ⊆ C) |
8 | 3, 7 | jctird 528 | . . 3 ⊢ ((F Fn B ∧ ran F ⊆ C) → ((G Fn A ∧ ran G ⊆ B) → ((F ∘ G) Fn A ∧ ran (F ∘ G) ⊆ C))) |
9 | 8 | imp 418 | . 2 ⊢ (((F Fn B ∧ ran F ⊆ C) ∧ (G Fn A ∧ ran G ⊆ B)) → ((F ∘ G) Fn A ∧ ran (F ∘ G) ⊆ C)) |
10 | df-f 4791 | . . 3 ⊢ (F:B–→C ↔ (F Fn B ∧ ran F ⊆ C)) | |
11 | df-f 4791 | . . 3 ⊢ (G:A–→B ↔ (G Fn A ∧ ran G ⊆ B)) | |
12 | 10, 11 | anbi12i 678 | . 2 ⊢ ((F:B–→C ∧ G:A–→B) ↔ ((F Fn B ∧ ran F ⊆ C) ∧ (G Fn A ∧ ran G ⊆ B))) |
13 | df-f 4791 | . 2 ⊢ ((F ∘ G):A–→C ↔ ((F ∘ G) Fn A ∧ ran (F ∘ G) ⊆ C)) | |
14 | 9, 12, 13 | 3imtr4i 257 | 1 ⊢ ((F:B–→C ∧ G:A–→B) → (F ∘ G):A–→C) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ⊆ wss 3257 ∘ ccom 4721 ran crn 4773 Fn wfn 4776 –→wf 4777 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 |
This theorem is referenced by: f1co 5264 foco 5279 enmap2lem5 6067 enmap1lem5 6073 |
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