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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.) |
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 2610 | . . 3 ⊢ 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉 | |
2 | funopsn.x | . . . 4 ⊢ 𝑋 ∈ V | |
3 | funopsn.y | . . . 4 ⊢ 𝑌 ∈ V | |
4 | 2, 3 | funopsn 6319 | . . 3 ⊢ ((Fun 〈𝑋, 𝑌〉 ∧ 〈𝑋, 𝑌〉 = 〈𝑋, 𝑌〉) → ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
5 | 1, 4 | mpan2 703 | . 2 ⊢ (Fun 〈𝑋, 𝑌〉 → ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
6 | vex 3176 | . . . . . 6 ⊢ 𝑎 ∈ V | |
7 | 6, 6 | funsn 5853 | . . . . 5 ⊢ Fun {〈𝑎, 𝑎〉} |
8 | funeq 5823 | . . . . 5 ⊢ (〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉} → (Fun 〈𝑋, 𝑌〉 ↔ Fun {〈𝑎, 𝑎〉})) | |
9 | 7, 8 | mpbiri 247 | . . . 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 1845 | . 2 ⊢ (∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉}) → Fun 〈𝑋, 𝑌〉) |
12 | 5, 11 | impbii 198 | 1 ⊢ (Fun 〈𝑋, 𝑌〉 ↔ ∃𝑎(𝑋 = {𝑎} ∧ 〈𝑋, 𝑌〉 = {〈𝑎, 𝑎〉})) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 {csn 4125 〈cop 4131 Fun wfun 5798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: funsndifnop 6321 fundmge2nop0 13129 |
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