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.) |
Ref | Expression |
---|---|
funop1 | ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq12 4342 | . . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑣, 𝑤〉) | |
2 | 1 | eqeq2d 2620 | . . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝐹 = 〈𝑥, 𝑦〉 ↔ 𝐹 = 〈𝑣, 𝑤〉)) |
3 | 2 | cbvex2v 2275 | . 2 ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 ↔ ∃𝑣∃𝑤 𝐹 = 〈𝑣, 𝑤〉) |
4 | vex 3176 | . . . . . . 7 ⊢ 𝑣 ∈ V | |
5 | vex 3176 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
6 | 4, 5 | funopsn 6319 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = 〈𝑣, 𝑤〉) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉})) |
7 | vex 3176 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
8 | opeq12 4342 | . . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑎〉) | |
9 | 8 | sneqd 4137 | . . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {〈𝑥, 𝑦〉} = {〈𝑎, 𝑎〉}) |
10 | 9 | eqeq2d 2620 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹 = {〈𝑥, 𝑦〉} ↔ 𝐹 = {〈𝑎, 𝑎〉})) |
11 | 7, 7, 10 | spc2ev 3274 | . . . . . . . 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 1845 | . . . . . 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 450 | . . . 4 ⊢ (𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
16 | vex 3176 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
17 | vex 3176 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
18 | 16, 17 | funsn 5853 | . . . . . 6 ⊢ Fun {〈𝑥, 𝑦〉} |
19 | funeq 5823 | . . . . . 6 ⊢ (𝐹 = {〈𝑥, 𝑦〉} → (Fun 𝐹 ↔ Fun {〈𝑥, 𝑦〉})) | |
20 | 18, 19 | mpbiri 247 | . . . . 5 ⊢ (𝐹 = {〈𝑥, 𝑦〉} → Fun 𝐹) |
21 | 20 | exlimivv 1847 | . . . 4 ⊢ (∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉} → Fun 𝐹) |
22 | 15, 21 | impbid1 214 | . . 3 ⊢ (𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
23 | 22 | exlimivv 1847 | . 2 ⊢ (∃𝑣∃𝑤 𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
24 | 3, 23 | sylbi 206 | 1 ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 {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: (None) |
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