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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > funop1 | Structured version Visualization version GIF version |
Description: A function is an ordered pair iff it is a singleton of an ordered pair. (Contributed by AV, 20-Sep-2020.) (Avoid depending on this detail.) |
Ref | Expression |
---|---|
funop1 | ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq12 4879 | . . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑣, 𝑤〉) | |
2 | 1 | eqeq2d 2745 | . . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝐹 = 〈𝑥, 𝑦〉 ↔ 𝐹 = 〈𝑣, 𝑤〉)) |
3 | 2 | cbvex2vw 2037 | . 2 ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 ↔ ∃𝑣∃𝑤 𝐹 = 〈𝑣, 𝑤〉) |
4 | vex 3481 | . . . . . . 7 ⊢ 𝑣 ∈ V | |
5 | vex 3481 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
6 | 4, 5 | funopsn 7167 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = 〈𝑣, 𝑤〉) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉})) |
7 | vex 3481 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
8 | opeq12 4879 | . . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑎〉) | |
9 | 8 | sneqd 4642 | . . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {〈𝑥, 𝑦〉} = {〈𝑎, 𝑎〉}) |
10 | 9 | eqeq2d 2745 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹 = {〈𝑥, 𝑦〉} ↔ 𝐹 = {〈𝑎, 𝑎〉})) |
11 | 7, 7, 10 | spc2ev 3606 | . . . . . . . 8 ⊢ (𝐹 = {〈𝑎, 𝑎〉} → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉}) |
12 | 11 | adantl 481 | . . . . . . 7 ⊢ ((𝑣 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉}) → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉}) |
13 | 12 | exlimiv 1927 | . . . . . 6 ⊢ (∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉}) → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉}) |
14 | 6, 13 | syl 17 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐹 = 〈𝑣, 𝑤〉) → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉}) |
15 | 14 | expcom 413 | . . . 4 ⊢ (𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
16 | vex 3481 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
17 | vex 3481 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
18 | 16, 17 | funsn 6620 | . . . . . 6 ⊢ Fun {〈𝑥, 𝑦〉} |
19 | funeq 6587 | . . . . . 6 ⊢ (𝐹 = {〈𝑥, 𝑦〉} → (Fun 𝐹 ↔ Fun {〈𝑥, 𝑦〉})) | |
20 | 18, 19 | mpbiri 258 | . . . . 5 ⊢ (𝐹 = {〈𝑥, 𝑦〉} → Fun 𝐹) |
21 | 20 | exlimivv 1929 | . . . 4 ⊢ (∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉} → Fun 𝐹) |
22 | 15, 21 | impbid1 225 | . . 3 ⊢ (𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
23 | 22 | exlimivv 1929 | . 2 ⊢ (∃𝑣∃𝑤 𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
24 | 3, 23 | sylbi 217 | 1 ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 {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: (None) |
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