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Mirrors > Home > NFE Home > Th. List > elimapw1 | GIF version |
Description: Membership in an image under a unit power class. (Contributed by set.mm contributors, 19-Feb-2015.) |
Ref | Expression |
---|---|
elimapw1 | ⊢ (A ∈ (B “ ℘1C) ↔ ∃x ∈ C 〈{x}, A〉 ∈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elima 4754 | . 2 ⊢ (A ∈ (B “ ℘1C) ↔ ∃t ∈ ℘1 CtBA) | |
2 | df-rex 2620 | . . . . 5 ⊢ (∃t ∈ ℘1 CtBA ↔ ∃t(t ∈ ℘1C ∧ tBA)) | |
3 | elpw1 4144 | . . . . . . . . 9 ⊢ (t ∈ ℘1C ↔ ∃x ∈ C t = {x}) | |
4 | 3 | anbi1i 676 | . . . . . . . 8 ⊢ ((t ∈ ℘1C ∧ tBA) ↔ (∃x ∈ C t = {x} ∧ tBA)) |
5 | r19.41v 2764 | . . . . . . . 8 ⊢ (∃x ∈ C (t = {x} ∧ tBA) ↔ (∃x ∈ C t = {x} ∧ tBA)) | |
6 | 4, 5 | bitr4i 243 | . . . . . . 7 ⊢ ((t ∈ ℘1C ∧ tBA) ↔ ∃x ∈ C (t = {x} ∧ tBA)) |
7 | 6 | exbii 1582 | . . . . . 6 ⊢ (∃t(t ∈ ℘1C ∧ tBA) ↔ ∃t∃x ∈ C (t = {x} ∧ tBA)) |
8 | rexcom4 2878 | . . . . . 6 ⊢ (∃x ∈ C ∃t(t = {x} ∧ tBA) ↔ ∃t∃x ∈ C (t = {x} ∧ tBA)) | |
9 | 7, 8 | bitr4i 243 | . . . . 5 ⊢ (∃t(t ∈ ℘1C ∧ tBA) ↔ ∃x ∈ C ∃t(t = {x} ∧ tBA)) |
10 | 2, 9 | bitri 240 | . . . 4 ⊢ (∃t ∈ ℘1 CtBA ↔ ∃x ∈ C ∃t(t = {x} ∧ tBA)) |
11 | snex 4111 | . . . . . 6 ⊢ {x} ∈ V | |
12 | breq1 4642 | . . . . . 6 ⊢ (t = {x} → (tBA ↔ {x}BA)) | |
13 | 11, 12 | ceqsexv 2894 | . . . . 5 ⊢ (∃t(t = {x} ∧ tBA) ↔ {x}BA) |
14 | 13 | rexbii 2639 | . . . 4 ⊢ (∃x ∈ C ∃t(t = {x} ∧ tBA) ↔ ∃x ∈ C {x}BA) |
15 | 10, 14 | bitri 240 | . . 3 ⊢ (∃t ∈ ℘1 CtBA ↔ ∃x ∈ C {x}BA) |
16 | df-br 4640 | . . . 4 ⊢ ({x}BA ↔ 〈{x}, A〉 ∈ B) | |
17 | 16 | rexbii 2639 | . . 3 ⊢ (∃x ∈ C {x}BA ↔ ∃x ∈ C 〈{x}, A〉 ∈ B) |
18 | 15, 17 | bitri 240 | . 2 ⊢ (∃t ∈ ℘1 CtBA ↔ ∃x ∈ C 〈{x}, A〉 ∈ B) |
19 | 1, 18 | bitri 240 | 1 ⊢ (A ∈ (B “ ℘1C) ↔ ∃x ∈ C 〈{x}, A〉 ∈ B) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 {csn 3737 ℘1cpw1 4135 〈cop 4561 class class class wbr 4639 “ cima 4722 |
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-br 4640 df-ima 4727 |
This theorem is referenced by: elimapw12 4945 elima1c 4947 elimapw11c 4948 otsnelsi3 5805 qsexg 5982 |
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