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 4808 | . . . 4 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → 〈𝑥, 𝑦〉 = 〈𝑣, 𝑤〉) | |
2 | 1 | eqeq2d 2835 | . . 3 ⊢ ((𝑥 = 𝑣 ∧ 𝑦 = 𝑤) → (𝐹 = 〈𝑥, 𝑦〉 ↔ 𝐹 = 〈𝑣, 𝑤〉)) |
3 | 2 | cbvex2vw 2047 | . 2 ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 ↔ ∃𝑣∃𝑤 𝐹 = 〈𝑣, 𝑤〉) |
4 | vex 3500 | . . . . . . 7 ⊢ 𝑣 ∈ V | |
5 | vex 3500 | . . . . . . 7 ⊢ 𝑤 ∈ V | |
6 | 4, 5 | funopsn 6913 | . . . . . 6 ⊢ ((Fun 𝐹 ∧ 𝐹 = 〈𝑣, 𝑤〉) → ∃𝑎(𝑣 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉})) |
7 | vex 3500 | . . . . . . . . 9 ⊢ 𝑎 ∈ V | |
8 | opeq12 4808 | . . . . . . . . . . 11 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → 〈𝑥, 𝑦〉 = 〈𝑎, 𝑎〉) | |
9 | 8 | sneqd 4582 | . . . . . . . . . 10 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → {〈𝑥, 𝑦〉} = {〈𝑎, 𝑎〉}) |
10 | 9 | eqeq2d 2835 | . . . . . . . . 9 ⊢ ((𝑥 = 𝑎 ∧ 𝑦 = 𝑎) → (𝐹 = {〈𝑥, 𝑦〉} ↔ 𝐹 = {〈𝑎, 𝑎〉})) |
11 | 7, 7, 10 | spc2ev 3611 | . . . . . . . 8 ⊢ (𝐹 = {〈𝑎, 𝑎〉} → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉}) |
12 | 11 | adantl 484 | . . . . . . 7 ⊢ ((𝑣 = {𝑎} ∧ 𝐹 = {〈𝑎, 𝑎〉}) → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉}) |
13 | 12 | exlimiv 1930 | . . . . . 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 416 | . . . 4 ⊢ (𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 → ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
16 | vex 3500 | . . . . . . 7 ⊢ 𝑥 ∈ V | |
17 | vex 3500 | . . . . . . 7 ⊢ 𝑦 ∈ V | |
18 | 16, 17 | funsn 6410 | . . . . . 6 ⊢ Fun {〈𝑥, 𝑦〉} |
19 | funeq 6378 | . . . . . 6 ⊢ (𝐹 = {〈𝑥, 𝑦〉} → (Fun 𝐹 ↔ Fun {〈𝑥, 𝑦〉})) | |
20 | 18, 19 | mpbiri 260 | . . . . 5 ⊢ (𝐹 = {〈𝑥, 𝑦〉} → Fun 𝐹) |
21 | 20 | exlimivv 1932 | . . . 4 ⊢ (∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉} → Fun 𝐹) |
22 | 15, 21 | impbid1 227 | . . 3 ⊢ (𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
23 | 22 | exlimivv 1932 | . 2 ⊢ (∃𝑣∃𝑤 𝐹 = 〈𝑣, 𝑤〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
24 | 3, 23 | sylbi 219 | 1 ⊢ (∃𝑥∃𝑦 𝐹 = 〈𝑥, 𝑦〉 → (Fun 𝐹 ↔ ∃𝑥∃𝑦 𝐹 = {〈𝑥, 𝑦〉})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∃wex 1779 {csn 4570 〈cop 4576 Fun wfun 6352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 |
This theorem is referenced by: (None) |
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