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Mirrors > Home > MPE Home > Th. List > fsn2 | Structured version Visualization version GIF version |
Description: A function that maps a singleton to a class is the singleton of an ordered pair. (Contributed by NM, 19-May-2004.) |
Ref | Expression |
---|---|
fsn2.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
fsn2 | ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsn2.1 | . . . . . 6 ⊢ 𝐴 ∈ V | |
2 | 1 | snid 4563 | . . . . 5 ⊢ 𝐴 ∈ {𝐴} |
3 | ffvelrn 6880 | . . . . 5 ⊢ ((𝐹:{𝐴}⟶𝐵 ∧ 𝐴 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵) | |
4 | 2, 3 | mpan2 691 | . . . 4 ⊢ (𝐹:{𝐴}⟶𝐵 → (𝐹‘𝐴) ∈ 𝐵) |
5 | ffn 6523 | . . . . 5 ⊢ (𝐹:{𝐴}⟶𝐵 → 𝐹 Fn {𝐴}) | |
6 | dffn3 6536 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} ↔ 𝐹:{𝐴}⟶ran 𝐹) | |
7 | 6 | biimpi 219 | . . . . . 6 ⊢ (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶ran 𝐹) |
8 | imadmrn 5924 | . . . . . . . . 9 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
9 | fndm 6459 | . . . . . . . . . 10 ⊢ (𝐹 Fn {𝐴} → dom 𝐹 = {𝐴}) | |
10 | 9 | imaeq2d 5914 | . . . . . . . . 9 ⊢ (𝐹 Fn {𝐴} → (𝐹 “ dom 𝐹) = (𝐹 “ {𝐴})) |
11 | 8, 10 | eqtr3id 2785 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → ran 𝐹 = (𝐹 “ {𝐴})) |
12 | fnsnfv 6768 | . . . . . . . . 9 ⊢ ((𝐹 Fn {𝐴} ∧ 𝐴 ∈ {𝐴}) → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) | |
13 | 2, 12 | mpan2 691 | . . . . . . . 8 ⊢ (𝐹 Fn {𝐴} → {(𝐹‘𝐴)} = (𝐹 “ {𝐴})) |
14 | 11, 13 | eqtr4d 2774 | . . . . . . 7 ⊢ (𝐹 Fn {𝐴} → ran 𝐹 = {(𝐹‘𝐴)}) |
15 | 14 | feq3d 6510 | . . . . . 6 ⊢ (𝐹 Fn {𝐴} → (𝐹:{𝐴}⟶ran 𝐹 ↔ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
16 | 7, 15 | mpbid 235 | . . . . 5 ⊢ (𝐹 Fn {𝐴} → 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) |
17 | 5, 16 | syl 17 | . . . 4 ⊢ (𝐹:{𝐴}⟶𝐵 → 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) |
18 | 4, 17 | jca 515 | . . 3 ⊢ (𝐹:{𝐴}⟶𝐵 → ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
19 | snssi 4707 | . . . 4 ⊢ ((𝐹‘𝐴) ∈ 𝐵 → {(𝐹‘𝐴)} ⊆ 𝐵) | |
20 | fss 6540 | . . . . 5 ⊢ ((𝐹:{𝐴}⟶{(𝐹‘𝐴)} ∧ {(𝐹‘𝐴)} ⊆ 𝐵) → 𝐹:{𝐴}⟶𝐵) | |
21 | 20 | ancoms 462 | . . . 4 ⊢ (({(𝐹‘𝐴)} ⊆ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) → 𝐹:{𝐴}⟶𝐵) |
22 | 19, 21 | sylan 583 | . . 3 ⊢ (((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) → 𝐹:{𝐴}⟶𝐵) |
23 | 18, 22 | impbii 212 | . 2 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)})) |
24 | fvex 6708 | . . . 4 ⊢ (𝐹‘𝐴) ∈ V | |
25 | 1, 24 | fsn 6928 | . . 3 ⊢ (𝐹:{𝐴}⟶{(𝐹‘𝐴)} ↔ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉}) |
26 | 25 | anbi2i 626 | . 2 ⊢ (((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹:{𝐴}⟶{(𝐹‘𝐴)}) ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
27 | 23, 26 | bitri 278 | 1 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ ((𝐹‘𝐴) ∈ 𝐵 ∧ 𝐹 = {〈𝐴, (𝐹‘𝐴)〉})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ⊆ wss 3853 {csn 4527 〈cop 4533 dom cdm 5536 ran crn 5537 “ cima 5539 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 |
This theorem is referenced by: fsn2g 6931 fnressn 6951 fressnfv 6953 mapsnconst 8551 elixpsn 8596 en1 8676 en1OLD 8677 mat1dimelbas 21322 0spth 28163 wlkl0 28404 ldepsnlinclem1 45462 ldepsnlinclem2 45463 0aryfvalel 45596 1arymaptf1 45604 |
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