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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 4881 | . . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑣, 𝑤〉) | |
2 | 1 | eqeq2d 2737 | . . 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 3466 | . . . . . . 7 ⊢ 𝑣 ∈ V | |
5 | vex 3466 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
6 | 4, 5 | funopsn 7162 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = 〈𝑣, 𝑤〉) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉})) |
7 | vex 3466 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
8 | opeq12 4881 | . . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑎〉) | |
9 | 8 | sneqd 4645 | . . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {〈𝑥, 𝑦〉} = {〈𝑎, 𝑎〉}) |
10 | 9 | eqeq2d 2737 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹 = {〈𝑥, 𝑦〉} ↔ 𝐹 = {〈𝑎, 𝑎〉})) |
11 | 7, 7, 10 | spc2ev 3593 | . . . . . . . 8 ⊢ (𝐹 = {〈𝑎, 𝑎〉} → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉}) |
12 | 11 | adantl 480 | . . . . . . 7 ⊢ ((𝑣 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉}) → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉}) |
13 | 12 | exlimiv 1926 | . . . . . 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 412 | . . . 4 ⊢ (𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
16 | vex 3466 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
17 | vex 3466 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
18 | 16, 17 | funsn 6612 | . . . . . 6 ⊢ Fun {〈𝑥, 𝑦〉} |
19 | funeq 6579 | . . . . . 6 ⊢ (𝐹 = {〈𝑥, 𝑦〉} → (Fun 𝐹 ↔ Fun {〈𝑥, 𝑦〉})) | |
20 | 18, 19 | mpbiri 257 | . . . . 5 ⊢ (𝐹 = {〈𝑥, 𝑦〉} → Fun 𝐹) |
21 | 20 | exlimivv 1928 | . . . 4 ⊢ (∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉} → Fun 𝐹) |
22 | 15, 21 | impbid1 224 | . . 3 ⊢ (𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
23 | 22 | exlimivv 1928 | . 2 ⊢ (∃𝑣∃𝑤 𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
24 | 3, 23 | sylbi 216 | 1 ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∃wex 1774 {csn 4633 〈cop 4639 Fun wfun 6548 |
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 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 |
This theorem is referenced by: (None) |
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