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Mirrors > Home > MPE Home > Th. List > fsng | Structured version Visualization version GIF version |
Description: A function maps a singleton to a singleton iff it is the singleton of an ordered pair. (Contributed by NM, 26-Oct-2012.) |
Ref | Expression |
---|---|
fsng | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4558 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | feq2d 6481 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝑏})) |
3 | opeq1 4784 | . . . . 5 ⊢ (𝑎 = 𝐴 → 〈𝑎, 𝑏〉 = 〈𝐴, 𝑏〉) | |
4 | 3 | sneqd 4560 | . . . 4 ⊢ (𝑎 = 𝐴 → {〈𝑎, 𝑏〉} = {〈𝐴, 𝑏〉}) |
5 | 4 | eqeq2d 2831 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {〈𝑎, 𝑏〉} ↔ 𝐹 = {〈𝐴, 𝑏〉})) |
6 | 2, 5 | bibi12d 348 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {〈𝑎, 𝑏〉}) ↔ (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {〈𝐴, 𝑏〉}))) |
7 | sneq 4558 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
8 | 7 | feq3d 6482 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵})) |
9 | opeq2 4785 | . . . . 5 ⊢ (𝑏 = 𝐵 → 〈𝐴, 𝑏〉 = 〈𝐴, 𝐵〉) | |
10 | 9 | sneqd 4560 | . . . 4 ⊢ (𝑏 = 𝐵 → {〈𝐴, 𝑏〉} = {〈𝐴, 𝐵〉}) |
11 | 10 | eqeq2d 2831 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {〈𝐴, 𝑏〉} ↔ 𝐹 = {〈𝐴, 𝐵〉})) |
12 | 8, 11 | bibi12d 348 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {〈𝐴, 𝑏〉}) ↔ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉}))) |
13 | vex 3484 | . . 3 ⊢ 𝑎 ∈ V | |
14 | vex 3484 | . . 3 ⊢ 𝑏 ∈ V | |
15 | 13, 14 | fsn 6878 | . 2 ⊢ (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {〈𝑎, 𝑏〉}) |
16 | 6, 12, 15 | vtocl2g 3559 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {〈𝐴, 𝐵〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 {csn 4548 〈cop 4554 ⟶wf 6332 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pr 5311 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-op 4555 df-br 5048 df-opab 5110 df-id 5441 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 |
This theorem is referenced by: xpsng 6882 ftpg 6899 mapsnd 8431 axdc3lem4 9856 fseq1p1m1 12966 cats1un 14063 intopsn 17842 efmnd1bas 18036 grp1inv 18185 symg1bas 18497 esumsnf 31325 bnj149 32149 rngosn3 35229 k0004lem3 40584 ovnovollem1 43023 mapsnop 44473 snlindsntorlem 44605 lmod1zr 44628 |
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