New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > rnoprab | GIF version |
Description: The range of an operation class abstraction. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Apr-2013.) (Contributed by set.mm contributors, 30-Aug-2004.) (Revised by set.mm contributors, 19-Apr-2013.) |
Ref | Expression |
---|---|
rnoprab | ⊢ ran {〈〈x, y〉, z〉 ∣ φ} = {z ∣ ∃x∃yφ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfoprab2 5558 | . . 3 ⊢ {〈〈x, y〉, z〉 ∣ φ} = {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} | |
2 | 1 | rneqi 4957 | . 2 ⊢ ran {〈〈x, y〉, z〉 ∣ φ} = ran {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} |
3 | rnopab 4967 | . 2 ⊢ ran {〈w, z〉 ∣ ∃x∃y(w = 〈x, y〉 ∧ φ)} = {z ∣ ∃w∃x∃y(w = 〈x, y〉 ∧ φ)} | |
4 | exrot3 1744 | . . . 4 ⊢ (∃w∃x∃y(w = 〈x, y〉 ∧ φ) ↔ ∃x∃y∃w(w = 〈x, y〉 ∧ φ)) | |
5 | 19.41v 1901 | . . . . . 6 ⊢ (∃w(w = 〈x, y〉 ∧ φ) ↔ (∃w w = 〈x, y〉 ∧ φ)) | |
6 | vex 2862 | . . . . . . . . . 10 ⊢ x ∈ V | |
7 | vex 2862 | . . . . . . . . . 10 ⊢ y ∈ V | |
8 | 6, 7 | opex 4588 | . . . . . . . . 9 ⊢ 〈x, y〉 ∈ V |
9 | 8 | isseti 2865 | . . . . . . . 8 ⊢ ∃w w = 〈x, y〉 |
10 | 9 | biantrur 492 | . . . . . . 7 ⊢ (φ ↔ (∃w w = 〈x, y〉 ∧ φ)) |
11 | 10 | bicomi 193 | . . . . . 6 ⊢ ((∃w w = 〈x, y〉 ∧ φ) ↔ φ) |
12 | 5, 11 | bitri 240 | . . . . 5 ⊢ (∃w(w = 〈x, y〉 ∧ φ) ↔ φ) |
13 | 12 | 2exbii 1583 | . . . 4 ⊢ (∃x∃y∃w(w = 〈x, y〉 ∧ φ) ↔ ∃x∃yφ) |
14 | 4, 13 | bitri 240 | . . 3 ⊢ (∃w∃x∃y(w = 〈x, y〉 ∧ φ) ↔ ∃x∃yφ) |
15 | 14 | abbii 2465 | . 2 ⊢ {z ∣ ∃w∃x∃y(w = 〈x, y〉 ∧ φ)} = {z ∣ ∃x∃yφ} |
16 | 2, 3, 15 | 3eqtri 2377 | 1 ⊢ ran {〈〈x, y〉, z〉 ∣ φ} = {z ∣ ∃x∃yφ} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 358 ∃wex 1541 = wceq 1642 {cab 2339 〈cop 4561 {copab 4622 ran crn 4773 {coprab 5527 |
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-rn 4786 df-oprab 5528 |
This theorem is referenced by: rnoprab2 5577 rnmpt2 5717 |
Copyright terms: Public domain | W3C validator |