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Mirrors > Home > NFE Home > Th. List > rnco | GIF version |
Description: The range of the composition of two classes. (Contributed by set.mm contributors, 12-Dec-2006.) |
Ref | Expression |
---|---|
rnco | ⊢ ran (A ∘ B) = ran (A ↾ ran B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brco 4883 | . . . . 5 ⊢ (x(A ∘ B)y ↔ ∃z(xBz ∧ zAy)) | |
2 | 1 | exbii 1582 | . . . 4 ⊢ (∃x x(A ∘ B)y ↔ ∃x∃z(xBz ∧ zAy)) |
3 | excom 1741 | . . . 4 ⊢ (∃x∃z(xBz ∧ zAy) ↔ ∃z∃x(xBz ∧ zAy)) | |
4 | ancom 437 | . . . . . . 7 ⊢ ((∃x xBz ∧ zAy) ↔ (zAy ∧ ∃x xBz)) | |
5 | 19.41v 1901 | . . . . . . 7 ⊢ (∃x(xBz ∧ zAy) ↔ (∃x xBz ∧ zAy)) | |
6 | elrn 4896 | . . . . . . . 8 ⊢ (z ∈ ran B ↔ ∃x xBz) | |
7 | 6 | anbi2i 675 | . . . . . . 7 ⊢ ((zAy ∧ z ∈ ran B) ↔ (zAy ∧ ∃x xBz)) |
8 | 4, 5, 7 | 3bitr4i 268 | . . . . . 6 ⊢ (∃x(xBz ∧ zAy) ↔ (zAy ∧ z ∈ ran B)) |
9 | brres 4949 | . . . . . 6 ⊢ (z(A ↾ ran B)y ↔ (zAy ∧ z ∈ ran B)) | |
10 | 8, 9 | bitr4i 243 | . . . . 5 ⊢ (∃x(xBz ∧ zAy) ↔ z(A ↾ ran B)y) |
11 | 10 | exbii 1582 | . . . 4 ⊢ (∃z∃x(xBz ∧ zAy) ↔ ∃z z(A ↾ ran B)y) |
12 | 2, 3, 11 | 3bitri 262 | . . 3 ⊢ (∃x x(A ∘ B)y ↔ ∃z z(A ↾ ran B)y) |
13 | elrn 4896 | . . 3 ⊢ (y ∈ ran (A ∘ B) ↔ ∃x x(A ∘ B)y) | |
14 | elrn 4896 | . . 3 ⊢ (y ∈ ran (A ↾ ran B) ↔ ∃z z(A ↾ ran B)y) | |
15 | 12, 13, 14 | 3bitr4i 268 | . 2 ⊢ (y ∈ ran (A ∘ B) ↔ y ∈ ran (A ↾ ran B)) |
16 | 15 | eqriv 2350 | 1 ⊢ ran (A ∘ B) = ran (A ↾ ran B) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 class class class wbr 4639 ∘ ccom 4721 ran crn 4773 ↾ cres 4774 |
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-xp 4784 df-rn 4786 df-res 4788 |
This theorem is referenced by: rnco2 5088 |
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