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Mirrors > Home > NFE Home > Th. List > fsng | GIF version |
Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by set.mm contributors, 26-Oct-2012.) |
Ref | Expression |
---|---|
fsng | ⊢ ((A ∈ C ∧ B ∈ D) → (F:{A}–→{B} ↔ F = {〈A, B〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 3744 | . . . 4 ⊢ (a = A → {a} = {A}) | |
2 | 1 | feq2d 5215 | . . 3 ⊢ (a = A → (F:{a}–→{b} ↔ F:{A}–→{b})) |
3 | opeq1 4578 | . . . . 5 ⊢ (a = A → 〈a, b〉 = 〈A, b〉) | |
4 | 3 | sneqd 3746 | . . . 4 ⊢ (a = A → {〈a, b〉} = {〈A, b〉}) |
5 | 4 | eqeq2d 2364 | . . 3 ⊢ (a = A → (F = {〈a, b〉} ↔ F = {〈A, b〉})) |
6 | 2, 5 | bibi12d 312 | . 2 ⊢ (a = A → ((F:{a}–→{b} ↔ F = {〈a, b〉}) ↔ (F:{A}–→{b} ↔ F = {〈A, b〉}))) |
7 | sneq 3744 | . . . 4 ⊢ (b = B → {b} = {B}) | |
8 | feq3 5212 | . . . 4 ⊢ ({b} = {B} → (F:{A}–→{b} ↔ F:{A}–→{B})) | |
9 | 7, 8 | syl 15 | . . 3 ⊢ (b = B → (F:{A}–→{b} ↔ F:{A}–→{B})) |
10 | opeq2 4579 | . . . . 5 ⊢ (b = B → 〈A, b〉 = 〈A, B〉) | |
11 | 10 | sneqd 3746 | . . . 4 ⊢ (b = B → {〈A, b〉} = {〈A, B〉}) |
12 | 11 | eqeq2d 2364 | . . 3 ⊢ (b = B → (F = {〈A, b〉} ↔ F = {〈A, B〉})) |
13 | 9, 12 | bibi12d 312 | . 2 ⊢ (b = B → ((F:{A}–→{b} ↔ F = {〈A, b〉}) ↔ (F:{A}–→{B} ↔ F = {〈A, B〉}))) |
14 | vex 2862 | . . 3 ⊢ a ∈ V | |
15 | vex 2862 | . . 3 ⊢ b ∈ V | |
16 | 14, 15 | fsn 5432 | . 2 ⊢ (F:{a}–→{b} ↔ F = {〈a, b〉}) |
17 | 6, 13, 16 | vtocl2g 2918 | 1 ⊢ ((A ∈ C ∧ B ∈ D) → (F:{A}–→{B} ↔ F = {〈A, B〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 {csn 3737 〈cop 4561 –→wf 4777 |
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-co 4726 df-ima 4727 df-id 4767 df-xp 4784 df-cnv 4785 df-rn 4786 df-dm 4787 df-fun 4789 df-fn 4790 df-f 4791 df-f1 4792 df-fo 4793 df-f1o 4794 |
This theorem is referenced by: (None) |
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