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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 2819 | . . 3 ⊢ 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉 | |
2 | funopsn.x | . . . 4 ⊢ 𝑋 ∈ V | |
3 | funopsn.y | . . . 4 ⊢ 𝑌 ∈ V | |
4 | 2, 3 | funopsn 6903 | . . 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 3496 | . . . . . 6 ⊢ 𝑎 ∈ V | |
7 | 6, 6 | funsn 6400 | . . . . 5 ⊢ Fun {〈𝑎, 𝑎〉} |
8 | funeq 6368 | . . . . 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 1925 | . 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 1531 ∃wex 1774 ∈ wcel 2108 Vcvv 3493 {csn 4559 〈cop 4565 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-ral 3141 df-rex 3142 df-reu 3143 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: funopdmsn 6905 funsndifnop 6906 |
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