New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > dmuni | GIF version |
Description: The domain of a union. Part of Exercise 8 of [Enderton] p. 41. (Contributed by set.mm contributors, 3-Feb-2004.) |
Ref | Expression |
---|---|
dmuni | ⊢ dom ∪A = ∪x ∈ A dom x |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni 3894 | . . . . . 6 ⊢ (〈y, z〉 ∈ ∪A ↔ ∃x(〈y, z〉 ∈ x ∧ x ∈ A)) | |
2 | 1 | exbii 1582 | . . . . 5 ⊢ (∃z〈y, z〉 ∈ ∪A ↔ ∃z∃x(〈y, z〉 ∈ x ∧ x ∈ A)) |
3 | excom 1741 | . . . . 5 ⊢ (∃z∃x(〈y, z〉 ∈ x ∧ x ∈ A) ↔ ∃x∃z(〈y, z〉 ∈ x ∧ x ∈ A)) | |
4 | 19.41v 1901 | . . . . . . 7 ⊢ (∃z(〈y, z〉 ∈ x ∧ x ∈ A) ↔ (∃z〈y, z〉 ∈ x ∧ x ∈ A)) | |
5 | ancom 437 | . . . . . . . . 9 ⊢ ((x ∈ A ∧ y ∈ dom x) ↔ (y ∈ dom x ∧ x ∈ A)) | |
6 | eldm2 4899 | . . . . . . . . . 10 ⊢ (y ∈ dom x ↔ ∃z〈y, z〉 ∈ x) | |
7 | 6 | anbi1i 676 | . . . . . . . . 9 ⊢ ((y ∈ dom x ∧ x ∈ A) ↔ (∃z〈y, z〉 ∈ x ∧ x ∈ A)) |
8 | 5, 7 | bitri 240 | . . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ dom x) ↔ (∃z〈y, z〉 ∈ x ∧ x ∈ A)) |
9 | 8 | bicomi 193 | . . . . . . 7 ⊢ ((∃z〈y, z〉 ∈ x ∧ x ∈ A) ↔ (x ∈ A ∧ y ∈ dom x)) |
10 | 4, 9 | bitri 240 | . . . . . 6 ⊢ (∃z(〈y, z〉 ∈ x ∧ x ∈ A) ↔ (x ∈ A ∧ y ∈ dom x)) |
11 | 10 | exbii 1582 | . . . . 5 ⊢ (∃x∃z(〈y, z〉 ∈ x ∧ x ∈ A) ↔ ∃x(x ∈ A ∧ y ∈ dom x)) |
12 | 2, 3, 11 | 3bitri 262 | . . . 4 ⊢ (∃z〈y, z〉 ∈ ∪A ↔ ∃x(x ∈ A ∧ y ∈ dom x)) |
13 | df-rex 2620 | . . . 4 ⊢ (∃x ∈ A y ∈ dom x ↔ ∃x(x ∈ A ∧ y ∈ dom x)) | |
14 | 12, 13 | bitr4i 243 | . . 3 ⊢ (∃z〈y, z〉 ∈ ∪A ↔ ∃x ∈ A y ∈ dom x) |
15 | eldm2 4899 | . . 3 ⊢ (y ∈ dom ∪A ↔ ∃z〈y, z〉 ∈ ∪A) | |
16 | eliun 3973 | . . 3 ⊢ (y ∈ ∪x ∈ A dom x ↔ ∃x ∈ A y ∈ dom x) | |
17 | 14, 15, 16 | 3bitr4i 268 | . 2 ⊢ (y ∈ dom ∪A ↔ y ∈ ∪x ∈ A dom x) |
18 | 17 | eqriv 2350 | 1 ⊢ dom ∪A = ∪x ∈ A dom x |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 ∪cuni 3891 ∪ciun 3969 〈cop 4561 dom cdm 4772 |
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-iun 3971 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-ima 4727 df-cnv 4785 df-rn 4786 df-dm 4787 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |