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Mirrors > Home > MPE Home > Th. List > funop | Structured version Visualization version GIF version |
Description: An ordered pair is a function iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) A function is a class of ordered pairs, so the fact that an ordered pair may sometimes be itself a function is an "accident" depending on the specific encoding of ordered pairs as classes (in set.mm, the Kuratowski encoding). A more meaningful statement is funsng 6409, as relsnopg 5658 is to relop 5704. (New usage is discouraged.) |
Ref | Expression |
---|---|
funopsn.x | ⊢ 𝑋 ∈ V |
funopsn.y | ⊢ 𝑌 ∈ V |
Ref | Expression |
---|---|
funop | ⊢ (Fun 〈𝑋, 𝑌〉 ↔ ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2736 | . . 3 ⊢ 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉 | |
2 | funopsn.x | . . . 4 ⊢ 𝑋 ∈ V | |
3 | funopsn.y | . . . 4 ⊢ 𝑌 ∈ V | |
4 | 2, 3 | funopsn 6941 | . . 3 ⊢ ((Fun 〈𝑋, 𝑌〉 ∧ 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉) → ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
5 | 1, 4 | mpan2 691 | . 2 ⊢ (Fun 〈𝑋, 𝑌〉 → ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
6 | vex 3402 | . . . . . 6 ⊢ 𝑎 ∈ V | |
7 | 6, 6 | funsn 6411 | . . . . 5 ⊢ Fun {〈𝑎, 𝑎〉} |
8 | funeq 6378 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉} → (Fun 〈𝑋, 𝑌〉 ↔ Fun {〈𝑎, 𝑎〉})) | |
9 | 7, 8 | mpbiri 261 | . . . 4 ⊢ (〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉} → Fun 〈𝑋, 𝑌〉) |
10 | 9 | adantl 485 | . . 3 ⊢ ((𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉}) → Fun 〈𝑋, 𝑌〉) |
11 | 10 | exlimiv 1938 | . 2 ⊢ (∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉}) → Fun 〈𝑋, 𝑌〉) |
12 | 5, 11 | impbii 212 | 1 ⊢ (Fun 〈𝑋, 𝑌〉 ↔ ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 Vcvv 3398 {csn 4527 〈cop 4533 Fun wfun 6352 |
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-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 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-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 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: funopdmsn 6943 funsndifnop 6944 |
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