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Mirrors > Home > NFE Home > Th. List > coass | GIF version |
Description: Associative law for class composition. Theorem 27 of [Suppes] p. 64. Also Exercise 21 of [Enderton] p. 53. Interestingly, this law holds for any classes whatsoever, not just functions or even relations. (Contributed by set.mm contributors, 27-Jan-1997.) |
Ref | Expression |
---|---|
coass | ⊢ ((A ∘ B) ∘ C) = (A ∘ (B ∘ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | excom 1741 | . . . . 5 ⊢ (∃w∃z((xCz ∧ zBw) ∧ wAy) ↔ ∃z∃w((xCz ∧ zBw) ∧ wAy)) | |
2 | anass 630 | . . . . . 6 ⊢ (((xCz ∧ zBw) ∧ wAy) ↔ (xCz ∧ (zBw ∧ wAy))) | |
3 | 2 | 2exbii 1583 | . . . . 5 ⊢ (∃z∃w((xCz ∧ zBw) ∧ wAy) ↔ ∃z∃w(xCz ∧ (zBw ∧ wAy))) |
4 | 1, 3 | bitr2i 241 | . . . 4 ⊢ (∃z∃w(xCz ∧ (zBw ∧ wAy)) ↔ ∃w∃z((xCz ∧ zBw) ∧ wAy)) |
5 | brco 4883 | . . . . . . 7 ⊢ (z(A ∘ B)y ↔ ∃w(zBw ∧ wAy)) | |
6 | 5 | anbi2i 675 | . . . . . 6 ⊢ ((xCz ∧ z(A ∘ B)y) ↔ (xCz ∧ ∃w(zBw ∧ wAy))) |
7 | 19.42v 1905 | . . . . . 6 ⊢ (∃w(xCz ∧ (zBw ∧ wAy)) ↔ (xCz ∧ ∃w(zBw ∧ wAy))) | |
8 | 6, 7 | bitr4i 243 | . . . . 5 ⊢ ((xCz ∧ z(A ∘ B)y) ↔ ∃w(xCz ∧ (zBw ∧ wAy))) |
9 | 8 | exbii 1582 | . . . 4 ⊢ (∃z(xCz ∧ z(A ∘ B)y) ↔ ∃z∃w(xCz ∧ (zBw ∧ wAy))) |
10 | brco 4883 | . . . . . . 7 ⊢ (x(B ∘ C)w ↔ ∃z(xCz ∧ zBw)) | |
11 | 10 | anbi1i 676 | . . . . . 6 ⊢ ((x(B ∘ C)w ∧ wAy) ↔ (∃z(xCz ∧ zBw) ∧ wAy)) |
12 | 19.41v 1901 | . . . . . 6 ⊢ (∃z((xCz ∧ zBw) ∧ wAy) ↔ (∃z(xCz ∧ zBw) ∧ wAy)) | |
13 | 11, 12 | bitr4i 243 | . . . . 5 ⊢ ((x(B ∘ C)w ∧ wAy) ↔ ∃z((xCz ∧ zBw) ∧ wAy)) |
14 | 13 | exbii 1582 | . . . 4 ⊢ (∃w(x(B ∘ C)w ∧ wAy) ↔ ∃w∃z((xCz ∧ zBw) ∧ wAy)) |
15 | 4, 9, 14 | 3bitr4i 268 | . . 3 ⊢ (∃z(xCz ∧ z(A ∘ B)y) ↔ ∃w(x(B ∘ C)w ∧ wAy)) |
16 | brco 4883 | . . 3 ⊢ (x((A ∘ B) ∘ C)y ↔ ∃z(xCz ∧ z(A ∘ B)y)) | |
17 | brco 4883 | . . 3 ⊢ (x(A ∘ (B ∘ C))y ↔ ∃w(x(B ∘ C)w ∧ wAy)) | |
18 | 15, 16, 17 | 3bitr4i 268 | . 2 ⊢ (x((A ∘ B) ∘ C)y ↔ x(A ∘ (B ∘ C))y) |
19 | 18 | eqbrriv 4851 | 1 ⊢ ((A ∘ B) ∘ C) = (A ∘ (B ∘ C)) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 class class class wbr 4639 ∘ ccom 4721 |
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 |
This theorem is referenced by: cnvpprod 5841 enmap2lem3 6065 enmap2lem5 6067 enmap1lem3 6071 enmap1lem5 6073 |
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