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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 4597 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴}) | |
2 | 1 | feq2d 6655 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝑏})) |
3 | opeq1 4831 | . . . . 5 ⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩) | |
4 | 3 | sneqd 4599 | . . . 4 ⊢ (𝑎 = 𝐴 → {⟨𝑎, 𝑏⟩} = {⟨𝐴, 𝑏⟩}) |
5 | 4 | eqeq2d 2744 | . . 3 ⊢ (𝑎 = 𝐴 → (𝐹 = {⟨𝑎, 𝑏⟩} ↔ 𝐹 = {⟨𝐴, 𝑏⟩})) |
6 | 2, 5 | bibi12d 346 | . 2 ⊢ (𝑎 = 𝐴 → ((𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩}) ↔ (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {⟨𝐴, 𝑏⟩}))) |
7 | sneq 4597 | . . . 4 ⊢ (𝑏 = 𝐵 → {𝑏} = {𝐵}) | |
8 | 7 | feq3d 6656 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹:{𝐴}⟶{𝑏} ↔ 𝐹:{𝐴}⟶{𝐵})) |
9 | opeq2 4832 | . . . . 5 ⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩) | |
10 | 9 | sneqd 4599 | . . . 4 ⊢ (𝑏 = 𝐵 → {⟨𝐴, 𝑏⟩} = {⟨𝐴, 𝐵⟩}) |
11 | 10 | eqeq2d 2744 | . . 3 ⊢ (𝑏 = 𝐵 → (𝐹 = {⟨𝐴, 𝑏⟩} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})) |
12 | 8, 11 | bibi12d 346 | . 2 ⊢ (𝑏 = 𝐵 → ((𝐹:{𝐴}⟶{𝑏} ↔ 𝐹 = {⟨𝐴, 𝑏⟩}) ↔ (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩}))) |
13 | vex 3448 | . . 3 ⊢ 𝑎 ∈ V | |
14 | vex 3448 | . . 3 ⊢ 𝑏 ∈ V | |
15 | 13, 14 | fsn 7082 | . 2 ⊢ (𝐹:{𝑎}⟶{𝑏} ↔ 𝐹 = {⟨𝑎, 𝑏⟩}) |
16 | 6, 12, 15 | vtocl2g 3530 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → (𝐹:{𝐴}⟶{𝐵} ↔ 𝐹 = {⟨𝐴, 𝐵⟩})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {csn 4587 ⟨cop 4593 ⟶wf 6493 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 |
This theorem is referenced by: xpsng 7086 ftpg 7103 mapsnd 8827 axdc3lem4 10394 fseq1p1m1 13521 cats1un 14615 intopsn 18514 efmnd1bas 18708 grp1inv 18860 symg1bas 19177 esumsnf 32720 bnj149 33544 rngosn3 36429 sticksstones9 40608 sticksstones11 40610 k0004lem3 42509 ovnovollem1 44983 mapsnop 46506 snlindsntorlem 46637 lmod1zr 46660 |
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