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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.) (Avoid depending on this detail.) |
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 2821 | . . 3 ⊢ 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉 | |
2 | funopsn.x | . . . 4 ⊢ 𝑋 ∈ V | |
3 | funopsn.y | . . . 4 ⊢ 𝑌 ∈ V | |
4 | 2, 3 | funopsn 6904 | . . 3 ⊢ ((Fun 〈𝑋, 𝑌〉 ∧ 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉) → ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
5 | 1, 4 | mpan2 689 | . 2 ⊢ (Fun 〈𝑋, 𝑌〉 → ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
6 | vex 3497 | . . . . . 6 ⊢ 𝑎 ∈ V | |
7 | 6, 6 | funsn 6401 | . . . . 5 ⊢ Fun {〈𝑎, 𝑎〉} |
8 | funeq 6369 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉} → (Fun 〈𝑋, 𝑌〉 ↔ Fun {〈𝑎, 𝑎〉})) | |
9 | 7, 8 | mpbiri 260 | . . . 4 ⊢ (〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉} → Fun 〈𝑋, 𝑌〉) |
10 | 9 | adantl 484 | . . 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 211 | 1 ⊢ (Fun 〈𝑋, 𝑌〉 ↔ ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 Vcvv 3494 {csn 4560 〈cop 4566 Fun wfun 6343 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: funopdmsn 6906 funsndifnop 6907 |
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