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Mirrors > Home > NFE Home > Th. List > brsnsi2 | GIF version |
Description: Binary relationship of an arbitrary set to a singleton in a singleton image. (Contributed by SF, 9-Mar-2015.) |
Ref | Expression |
---|---|
brsnsi1.1 | ⊢ A ∈ V |
Ref | Expression |
---|---|
brsnsi2 | ⊢ (B SI R{A} ↔ ∃x(B = {x} ∧ xRA)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brsi 4761 | . 2 ⊢ (B SI R{A} ↔ ∃x∃y(B = {x} ∧ {A} = {y} ∧ xRy)) | |
2 | 3anass 938 | . . . . 5 ⊢ ((B = {x} ∧ {A} = {y} ∧ xRy) ↔ (B = {x} ∧ ({A} = {y} ∧ xRy))) | |
3 | 2 | exbii 1582 | . . . 4 ⊢ (∃y(B = {x} ∧ {A} = {y} ∧ xRy) ↔ ∃y(B = {x} ∧ ({A} = {y} ∧ xRy))) |
4 | 19.42v 1905 | . . . . 5 ⊢ (∃y(B = {x} ∧ ({A} = {y} ∧ xRy)) ↔ (B = {x} ∧ ∃y({A} = {y} ∧ xRy))) | |
5 | brsnsi1.1 | . . . . . . . . . . 11 ⊢ A ∈ V | |
6 | 5 | sneqb 3876 | . . . . . . . . . 10 ⊢ ({A} = {y} ↔ A = y) |
7 | eqcom 2355 | . . . . . . . . . 10 ⊢ (A = y ↔ y = A) | |
8 | 6, 7 | bitri 240 | . . . . . . . . 9 ⊢ ({A} = {y} ↔ y = A) |
9 | 8 | anbi1i 676 | . . . . . . . 8 ⊢ (({A} = {y} ∧ xRy) ↔ (y = A ∧ xRy)) |
10 | 9 | exbii 1582 | . . . . . . 7 ⊢ (∃y({A} = {y} ∧ xRy) ↔ ∃y(y = A ∧ xRy)) |
11 | breq2 4643 | . . . . . . . 8 ⊢ (y = A → (xRy ↔ xRA)) | |
12 | 5, 11 | ceqsexv 2894 | . . . . . . 7 ⊢ (∃y(y = A ∧ xRy) ↔ xRA) |
13 | 10, 12 | bitri 240 | . . . . . 6 ⊢ (∃y({A} = {y} ∧ xRy) ↔ xRA) |
14 | 13 | anbi2i 675 | . . . . 5 ⊢ ((B = {x} ∧ ∃y({A} = {y} ∧ xRy)) ↔ (B = {x} ∧ xRA)) |
15 | 4, 14 | bitri 240 | . . . 4 ⊢ (∃y(B = {x} ∧ ({A} = {y} ∧ xRy)) ↔ (B = {x} ∧ xRA)) |
16 | 3, 15 | bitri 240 | . . 3 ⊢ (∃y(B = {x} ∧ {A} = {y} ∧ xRy) ↔ (B = {x} ∧ xRA)) |
17 | 16 | exbii 1582 | . 2 ⊢ (∃x∃y(B = {x} ∧ {A} = {y} ∧ xRy) ↔ ∃x(B = {x} ∧ xRA)) |
18 | 1, 17 | bitri 240 | 1 ⊢ (B SI R{A} ↔ ∃x(B = {x} ∧ xRA)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∧ w3a 934 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 {csn 3737 class class class wbr 4639 SI csi 4720 |
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-si 4728 |
This theorem is referenced by: brimage 5793 enpw1lem1 6061 nmembers1lem1 6268 nchoicelem11 6299 |
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