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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 6618, as relsnopg 5815 is to relop 5863. (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 2734 | . . 3 ⊢ 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉 | |
2 | funopsn.x | . . . 4 ⊢ 𝑋 ∈ V | |
3 | funopsn.y | . . . 4 ⊢ 𝑌 ∈ V | |
4 | 2, 3 | funopsn 7167 | . . 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 3481 | . . . . . 6 ⊢ 𝑎 ∈ V | |
7 | 6, 6 | funsn 6620 | . . . . 5 ⊢ Fun {〈𝑎, 𝑎〉} |
8 | funeq 6587 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉} → (Fun 〈𝑋, 𝑌〉 ↔ Fun {〈𝑎, 𝑎〉})) | |
9 | 7, 8 | mpbiri 258 | . . . 4 ⊢ (〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉} → Fun 〈𝑋, 𝑌〉) |
10 | 9 | adantl 481 | . . 3 ⊢ ((𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉}) → Fun 〈𝑋, 𝑌〉) |
11 | 10 | exlimiv 1927 | . 2 ⊢ (∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉}) → Fun 〈𝑋, 𝑌〉) |
12 | 5, 11 | impbii 209 | 1 ⊢ (Fun 〈𝑋, 𝑌〉 ↔ ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 Vcvv 3477 {csn 4630 〈cop 4636 Fun wfun 6556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 |
This theorem is referenced by: funopdmsn 7169 funsndifnop 7170 |
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