New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > dmsnopg | GIF version |
Description: The domain of a singleton of an ordered pair is the singleton of the first member. (Contributed by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
dmsnopg | ⊢ (B ∈ V → dom {〈A, B〉} = {A}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 4579 | . . . . 5 ⊢ (y = B → 〈A, y〉 = 〈A, B〉) | |
2 | 1 | sneqd 3746 | . . . 4 ⊢ (y = B → {〈A, y〉} = {〈A, B〉}) |
3 | 2 | dmeqd 4909 | . . 3 ⊢ (y = B → dom {〈A, y〉} = dom {〈A, B〉}) |
4 | 3 | eqeq1d 2361 | . 2 ⊢ (y = B → (dom {〈A, y〉} = {A} ↔ dom {〈A, B〉} = {A})) |
5 | df-br 4640 | . . . . . . 7 ⊢ (x{〈A, y〉}z ↔ 〈x, z〉 ∈ {〈A, y〉}) | |
6 | vex 2862 | . . . . . . . . 9 ⊢ x ∈ V | |
7 | vex 2862 | . . . . . . . . 9 ⊢ z ∈ V | |
8 | 6, 7 | opex 4588 | . . . . . . . 8 ⊢ 〈x, z〉 ∈ V |
9 | 8 | elsnc 3756 | . . . . . . 7 ⊢ (〈x, z〉 ∈ {〈A, y〉} ↔ 〈x, z〉 = 〈A, y〉) |
10 | opth 4602 | . . . . . . . 8 ⊢ (〈x, z〉 = 〈A, y〉 ↔ (x = A ∧ z = y)) | |
11 | ancom 437 | . . . . . . . 8 ⊢ ((x = A ∧ z = y) ↔ (z = y ∧ x = A)) | |
12 | 10, 11 | bitri 240 | . . . . . . 7 ⊢ (〈x, z〉 = 〈A, y〉 ↔ (z = y ∧ x = A)) |
13 | 5, 9, 12 | 3bitri 262 | . . . . . 6 ⊢ (x{〈A, y〉}z ↔ (z = y ∧ x = A)) |
14 | 13 | exbii 1582 | . . . . 5 ⊢ (∃z x{〈A, y〉}z ↔ ∃z(z = y ∧ x = A)) |
15 | vex 2862 | . . . . . 6 ⊢ y ∈ V | |
16 | biidd 228 | . . . . . 6 ⊢ (z = y → (x = A ↔ x = A)) | |
17 | 15, 16 | ceqsexv 2894 | . . . . 5 ⊢ (∃z(z = y ∧ x = A) ↔ x = A) |
18 | 14, 17 | bitri 240 | . . . 4 ⊢ (∃z x{〈A, y〉}z ↔ x = A) |
19 | eldm 4898 | . . . 4 ⊢ (x ∈ dom {〈A, y〉} ↔ ∃z x{〈A, y〉}z) | |
20 | elsn 3748 | . . . 4 ⊢ (x ∈ {A} ↔ x = A) | |
21 | 18, 19, 20 | 3bitr4i 268 | . . 3 ⊢ (x ∈ dom {〈A, y〉} ↔ x ∈ {A}) |
22 | 21 | eqriv 2350 | . 2 ⊢ dom {〈A, y〉} = {A} |
23 | 4, 22 | vtoclg 2914 | 1 ⊢ (B ∈ V → dom {〈A, B〉} = {A}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 {csn 3737 〈cop 4561 class class class wbr 4639 dom cdm 4772 |
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-ima 4727 df-cnv 4785 df-rn 4786 df-dm 4787 |
This theorem is referenced by: dmsnopss 5067 dmpropg 5068 dmsnop 5069 funprg 5149 funprgOLD 5150 |
Copyright terms: Public domain | W3C validator |