New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > dmcoss | GIF version |
Description: Domain of a composition. Theorem 21 of [Suppes] p. 63. (The proof was shortened by Andrew Salmon, 27-Aug-2011.) (Contributed by set.mm contributors, 19-Mar-1998.) (Revised by set.mm contributors, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmcoss | ⊢ dom (A ∘ B) ⊆ dom B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brco 4884 | . . . . . 6 ⊢ (x(A ∘ B)y ↔ ∃z(xBz ∧ zAy)) | |
2 | 1 | exbii 1582 | . . . . 5 ⊢ (∃y x(A ∘ B)y ↔ ∃y∃z(xBz ∧ zAy)) |
3 | excom 1741 | . . . . 5 ⊢ (∃y∃z(xBz ∧ zAy) ↔ ∃z∃y(xBz ∧ zAy)) | |
4 | 2, 3 | bitri 240 | . . . 4 ⊢ (∃y x(A ∘ B)y ↔ ∃z∃y(xBz ∧ zAy)) |
5 | simpl 443 | . . . . . 6 ⊢ ((xBz ∧ zAy) → xBz) | |
6 | 5 | exlimiv 1634 | . . . . 5 ⊢ (∃y(xBz ∧ zAy) → xBz) |
7 | 6 | eximi 1576 | . . . 4 ⊢ (∃z∃y(xBz ∧ zAy) → ∃z xBz) |
8 | 4, 7 | sylbi 187 | . . 3 ⊢ (∃y x(A ∘ B)y → ∃z xBz) |
9 | eldm 4899 | . . 3 ⊢ (x ∈ dom (A ∘ B) ↔ ∃y x(A ∘ B)y) | |
10 | eldm 4899 | . . 3 ⊢ (x ∈ dom B ↔ ∃z xBz) | |
11 | 8, 9, 10 | 3imtr4i 257 | . 2 ⊢ (x ∈ dom (A ∘ B) → x ∈ dom B) |
12 | 11 | ssriv 3278 | 1 ⊢ dom (A ∘ B) ⊆ dom B |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 ∈ wcel 1710 ⊆ wss 3258 class class class wbr 4640 ∘ ccom 4722 dom cdm 4773 |
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 4079 ax-xp 4080 ax-cnv 4081 ax-1c 4082 ax-sset 4083 ax-si 4084 ax-ins2 4085 ax-ins3 4086 ax-typlower 4087 ax-sn 4088 |
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 2479 df-ne 2519 df-ral 2620 df-rex 2621 df-reu 2622 df-rmo 2623 df-rab 2624 df-v 2862 df-sbc 3048 df-nin 3212 df-compl 3213 df-in 3214 df-un 3215 df-dif 3216 df-symdif 3217 df-ss 3260 df-pss 3262 df-nul 3552 df-if 3664 df-pw 3725 df-sn 3742 df-pr 3743 df-uni 3893 df-int 3928 df-opk 4059 df-1c 4137 df-pw1 4138 df-uni1 4139 df-xpk 4186 df-cnvk 4187 df-ins2k 4188 df-ins3k 4189 df-imak 4190 df-cok 4191 df-p6 4192 df-sik 4193 df-ssetk 4194 df-imagek 4195 df-idk 4196 df-iota 4340 df-0c 4378 df-addc 4379 df-nnc 4380 df-fin 4381 df-lefin 4441 df-ltfin 4442 df-ncfin 4443 df-tfin 4444 df-evenfin 4445 df-oddfin 4446 df-sfin 4447 df-spfin 4448 df-phi 4566 df-op 4567 df-proj1 4568 df-proj2 4569 df-opab 4624 df-br 4641 df-co 4727 df-ima 4728 df-cnv 4786 df-rn 4787 df-dm 4788 |
This theorem is referenced by: rncoss 4973 dmcosseq 4974 txpcofun 5804 |
Copyright terms: Public domain | W3C validator |